I am reading about the LMM with lognormal-mixture dynamics. Consider the following dynamics for the forward rate $F_{i}(t)$ fixing at $T_{i-1}$ and paying at $T_i$: \begin{align} dF_{i}(t) = (F_i (t) +\gamma) \sigma_i^I dt \ , \end{align} where $\gamma$ is a constant shift and $\sigma_i^I$ is a random variable for the volatility drawn at time $0^+$ from a discrete or continuous distribution (the superscript $I$ denotes the state). These dynamics have the convenience that the price of a European call/put simply is the expectation of Black-Scholes prices over all possible scenarios for the volatility, and one could quite easily calibrate the states on the market smile of the caplet.
In an article by Piterbarg, "Mixture of Models: A Simple Recipe for a... Hangover?", it is argued that such a dynamics cannot be used for the pricing of exotics (one should for example first transform the dynamics into a local vol function).
However, I fail to understand his argument, especially on page 4. The author takes as an example a simple compounded put option with two states {1, 2} for the volatility. He then argues that when time passes by and if the market doesn't change, the continuation value $H(S)$ at time $T_1$ (first exercise date) will be calculated under the same mixture as follows: \begin{align} H(S) = p_1 E^1_{T_1}\{(K_2 - S_{T_2})^+ |S_{T_1} = S \} + p_2 E^2_{T_1}\{(K_2 - S_{T_2})^+ |S_{T_1} = S \} \ . \end{align} First of all, what I don't understand is that when a book of exotics is marked-to-market on a daily basis, even if the market doesn't change significantly, a new calibration procedure could lead to different calibrated states simply because the time to maturity changes (i.e. we get closer to $T_1$ and $T_2$ in the example above).
Does anybody has a clear explanation (either intuitively or mathematically) on why such a models can or can't be used in practice? Intuitively I would think that such a model would perform at least better than a simple LMM (only one state with probability 1) when it comes to hedging the smile. I understand that theoretically the model isn't very realistic, but I am looking at the problem from a practitioner's point of view. Any insights on the problematic would be useful.