Reduction-lattice conjecture for Weyl modules

Let $\mathbb F$ be an algebraically closed field of positive characteristic, let $\mathbb A$ be the coefficient ring used for reduction modulo the characteristic, and let $W^0(\gb\omega)$ be a characteristic-zero Weyl module with an $\ell$-highest-weight vector $v$. Set $L=U(\tlie g)_\mathbb A v$, let $\gb\omega$ reduce to $\overline{\gb\omega}$, and write $L_\mathbb F$ for the reduced lattice; for a factorization $\gb\omega=\prod_{j=1}^m\gb\omega_{\lambda,a_j}$, let $L'$ be the tensor product of the corresponding factor lattices. Reduction-lattice conjecture. In the notation above,

$$W(\overline{\gb\omega})\cong L_\mathbb F.$$

Moreover, if $\bar a_i\ne\bar a_j$ for $i\ne j$, then $L=L'$. This statement is presented as a corollary-like assertion in the proof of the reduction construction, but the supplied material gives no status evidence beyond the assertion itself.