Thomsen's comparison conjecture for the MVdK and KLT splittings

Let $G=\operatorname{GL}_n$, let $P$ be a parabolic subgroup containing the Borel subgroup $B$, and let $\mathfrak{u}_P$ be the nilpotent radical of the Lie algebra of $P$. Consider the cotangent bundles $G\times^B\mathfrak{u}_P$ and $G\times^P\mathfrak{u}_P$, together with the MVdK splitting and the KLT splitting. Thomsen's conjecture. The pushforward to $G\times^P\mathfrak{u}_P$ of the splitting of $G\times^B\mathfrak{u}_P$ induced by the MVdK splitting is the top-degree component of the KLT splitting. This compares two Frobenius splittings arising in the study of cotangent bundles of flag varieties; the supplied text does not indicate whether the assertion has been proved or remains open.