Forni's equidistribution conjecture for Teichmüller geodesic translates

Let $g\geq 1$, let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a partition of $2g-2$, let $\mathcal H_1(\alpha)$ be the space of unit-area surfaces in the corresponding stratum of abelian differentials, and let $x\in\mathcal H_1(\alpha)$. For an interval $I\subset\mathbb R$, write

$$\mu_I^x=|I|^{-1}\int_I\delta_{u_sx}\operatorname{d}\!s.$$

Here $a_t$ denotes the diagonal subgroup of ${\rm SL}_2(\mathbb R)$, and an affine invariant measure is an ${\rm SL}_2(\mathbb R)$-invariant probability measure supported on an affine invariant manifold. Forni's conjecture. The limit

$$\lim_{t\to\infty}a_t\mu^{x}_{[0,1]}$$

exists in the weak$^*$ sense and is equal to an affine invariant measure with $x$ in its support. The conjecture would give an effective or pointwise equidistribution statement complementing the existing averaged convergence theorem, while examples show that the unipotent averages themselves need not converge or may converge to a non-ergodic measure.