LMP conjecture on Frobenius splitting of the diagonal blow-up

Let $k$ be an algebraically closed field of positive characteristic $p$. Let $V$ be a rational projective homogeneous space, set $X=V\times V$, and let $\widetilde{X}$ be the blow-up of the diagonal $\Delta$ in $X$. Let $E$ be the exceptional divisor. LMP conjecture. The variety $\widetilde{X}$ is Frobenius split compatibly with the exceptional divisor $E$.

This conjecture, introduced by V. Lakshmibai, V. B. Mehta, and A. J. Parameswaran, is presented as implying Wahl's conjecture in positive characteristic. The source does not state whether the LMP conjecture itself has been resolved.