Non-reduced SNC divisor spectral sequence degeneration conjecture

Let $X$ be a Kähler manifold and let $D=\sum D_i$ be a reduced simple normal crossings divisor such that every intersection $D_{i_0}\cap\cdots\cap D_{i_p}$ is irreducible. Let

$$D_{(r)}=\sum r_iD_i$$

be an effective non-reduced divisor with simple normal crossings support, and let $\Delta=\Delta(D_{(r)})$ be its dual simplicial complex. Non-reduced SNC degeneration conjecture. The spectral sequence constructed for the structure sheaf $\mathcal O_{D_{(r)}}$ degenerates at the second page, and hence

$$H^i(D,\mathcal O_{D_{(r)}})\cong\bigoplus_{p+q=i}H^p(\Delta,\mathcal H^q(\mathcal O_{(r)})).$$

For reduced simple normal crossings divisors, analogous degeneration and decomposition results follow from the preceding theorem; the conjecture asks for the corresponding statement in the non-reduced case.