There are two ways to answer your question. One is direct and has less depth to it, the other is more indirect and has a lot of depth to it. I will begin with the indirect one because it has wide ranging applications beyond finance or economics and because it should serve as a warning to journal editors and so forth. Also, it covers an area of statistics that everyone but statisticians have forgotten about.
Although the idea of a statistic is rather old, the field of statistics is rather new. It is probably the newest or nearly newest of all fields. Aeronautics is older. Genetics is older. It opened up a ton of practical questions that it took time to solve. It has to do with how the field defined as statistic. A statistic is any function of the data. This means that almost every statistic is useless as there are uncountably many functions.
This led to a process to decide which statistics to keep and which to discard. This created unexpected results. If you are finding poor estimators as a result of your theory, then there is a good chance, you are doing it wrong. In the defense of finance, it has been struggling with this since Mandelbrot published the first empirical refutation of mean-variance finance. It tried to solve it in the 1960's but a couple of things got in the way. The first was the use of punch card technology. Even if work by Eugene Fama or Mandelbrot were correct, it would have resulted in problems that would take decades to solve. The second was that there was no reason for them to be correct. There was no theory behind the observations.
The unexpected result, in searching for a statistic, was that all Bayesian statistics were admissible. This was surprising because it was proved with Frequentist axioms. It also found that all other statistics were valid to the extent they either mapped to a Bayesian measure in a particular case or at the limit. It provides a test, however. If you can stochastically dominate a measure, then you drop that measure. If you hunting for accurate measurements that work, then something deeper is going on and you are missing it.
The more direct answer is that the distribution of returns, for the Markowtiz model to be correct, have to have certain properties. The first is that there needs to be a mean in order to have an expectation in the first place. Most standard distributions have a mean, but not all do. The Cauchy distribution and, in general, the Paretian distributions of Mandelbrot's article do not. The Cauchy distribution is $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(x-\mu)^2}.$$
The second is that if a mean exists, a covariance matrix needs to exist. Not all distributions with a variance have a covariance in its multivariate form. The hyperbolic secant distribution is an example of that. It is $$sech\left(\frac{x-\mu}{\sigma}\right).$$
There have been attempts to use both in empirical finance. If either of those distributions are present in the likelihood function, then mean-variance finance is indefensible. The former is problematic because you cannot form an expectation on your returns in the first place. They are excluded by the laws of general summation. The second is a bit more subtle because if the second is present none of the assets can be independent, but none of them can covary either. They can comove, but not covary. It creates a very ugly issue.
There is a paper that derives the distribution of returns at https://ssrn.com/abstract=2828744. It shows that there are many distributions that can be present. The logic of the paper is that returns are not data, rather, prices are data. Returns are transformations of data. In particular, they are the ratio of jointly distributed variables, a present value and a future value. The distribution depends upon the rules in use to create the prices. As a result, stocks have different returns than antiques because the auction process is different.
As it happens, all distributions for equity securities include some mixture of a transformation of the Cauchy distribution. Because the distributions involved lack a sufficient statistic, any point estimator has to lose information, so no non-Bayesian solution exists for projective problems (such as choosing an allocation), and should be avoided for inferential questions if possible. You cannot avoid them in your true hypothesis is a sharp null hypothesis as there is no good Bayesian solution for sharp null hypotheses.
A population test of the paper can be found at https://ssrn.com/abstract=2653151
There are also papers to replace the method of pricing options and the rules of econometrics. Papers to create optimal portfolios and to extend stochastic calculus are in process. The distributions paper will be presented at the Southerwestern Finance Association Conference in March.
Some things will have to change. You cannot make an assumption of i.i.d. variables, for example. The entire discussion of the Solow convergence will have to change in economics and so the core of the whole discussion of capital, physical, financial and human.
A lot of focus will end up on the scale parameter. In the Cauchy distribution, there is no covariance matrix. If you had a one asset portfolio, denoted $a$, then it may have a scale parameter $\gamma_a$. If you switch to a two asset portfolio you do not get two scale parameters, let alone a covariance style matrix. Instead you get a new scale parameter $\gamma_{ab}$. If you got fancy and used a vector process, all the vectors would jointly share a scale parameter $\gamma_v$. Taking the logarithm brings you to the hyperbolic secant distribution and so no gain is had. It also has no covariance matrix, but OLS does. OLS would be measuring something that does not exist.
The headaches are just starting.