Coming back to my own question after I replicated the paper successfully for my thesis, where I found that my resulting SDF is always strictly positive and hovering around the value 1, just as expected given the formulation. Then, I also looked at their data and code and realized that this formulation is maybe just one way to "enforce" No-Arbitrage ...
There are two common solutions.
You exclude from your universe stocks whose history is too short (i.e. only recently went public). Sometimes it breaks your heart to do that, because you really like the stock. (If the time series is long enough, but still has small gaps, there are ways to "fix" the covariance matrix.)
You use models to predict the ...
I am not a math guy but I tried to plot the efficient frontier using the linear combination between the global min var portfolio and another efficient portfolio and I have this result:
Points represent efficient portfolios from variance minimization given a target return
The linear combination looks more like an approximation