Convergence of expanding twist-torus measures for Veech surfaces

Let $(M,\omega)$ be a horizontally periodic Veech surface, let $\mathcal{M}=\overline{\mathrm{SL}_2(\mathbb{R})\cdot\mathbb{T}(\omega)}$, and let $\mu_\mathbb{T}$ be a fully supported Lebesgue probability measure on $\mathbb{T}(\omega)$. Assume that a pseudo-Anosov element of $\mathrm{Aff}^+(M,\omega)$ acts as the identity matrix on $\mathrm{Cyl}^0(\omega)$. Let $\mu_\mathcal{M}$ be the unique $\mathrm{SL}_2(\mathbb{R})$-ergodic probability measure fully supported on $\mathcal{M}$. Convergence conjecture. Then

$$(g_t)_*\mu_\mathbb{T}\longrightarrow\mu_\mathcal{M}\quad\text{as }t\rightarrow\infty.$$

The preceding theorem gives full support for every weak-$*$ limit under the stated hypothesis, and the conjecture predicts uniqueness of the limiting measure and convergence of the entire family. It is motivated by an affirmative answer to the cited problem for Veech surfaces satisfying the pseudo-Anosov identity hypothesis, but the statement remains open in the source.