Cohomological vanishing conjecture for dual complexes of rational singularities

Let $f:X\to Y$ be a good resolution of an isolated rational singularity $o\in Y$, and let the exceptional divisor be

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

with effective simple normal crossings support. Let $\Delta$ denote the dual complex of the exceptional divisor. Cohomological vanishing conjecture. One has

$$H^i(\Delta,\mathbb C)=0\qquad\text{for all }i>0.$$

This is proposed as a weaker assertion than contractibility of the dual complex for isolated rational singularities. It would follow from the stronger conjectural relationship between the cohomology of the dual complex and that of the non-reduced exceptional divisor, while the general case remains open.