Semiregularity conjecture for twisted coherent sheaves

Let $\pi:{{{{{{{{\mathcal M}}}}}}}}\rightarrow S$ be a deformation of a smooth complex projective variety $M_0$ over a smooth germ $(S,0)$, and write $M_s=\pi^{-1}(s)$. Let ${{{{{{{{\mathcal B}}}}}}}}$ be a semiregular rank-$r$ coherent sheaf on $M_0$, twisted by a cocycle with coefficients in $\mu_r$. Assume that, for every $p$, the class $ch_p({{{{{{{{\mathcal B}}}}}}}})$ extends to a horizontal section of $R^{2p}\pi_*\mathbb Q$ lying in the direct summand $R^p\pi_*\Omega^p_\pi$ under the Hodge decomposition. Semiregularity conjecture. The sheaf ${{{{{{{{\mathcal B}}}}}}}}$ extends to a twisted coherent sheaf over $\pi^{-1}(U)$ for some open analytic neighborhood $U$ of $0$ in $S$. This is a twisted-sheaf analogue of a semiregularity theorem; the source says that it should follow from existing work, so the status of this formulation is left open here.