Vanishing conjecture for log-pluricanonical sheaves with standard coefficients

Let $(X,\Delta)$ be a proper slc pair over a field of characteristic $0$ such that $K_X+\Delta$ is ample. Fix an integer $m\geq 2$, and suppose that the coefficients of $\Delta$ satisfy

$$\operatorname{coeff}\Delta\subset \left\{\tfrac12,\tfrac23,\tfrac34,\dots,1\right\}\cup \left[1-\tfrac1m,1\right].$$

Vanishing conjecture. The higher cohomology of the $m$-th reflexive log-pluricanonical sheaf vanishes:

$$H^i\bigl(X,\omega_X^{[m]}(\lfloor m\Delta\rfloor)\bigr)=0\qquad\text{for all }i>0.$$

This vanishing would extend the known surface results to proper slc pairs in arbitrary dimension and, via cohomology and base change, would yield invariance and base-change results for log-plurigenera in stable families. The statement is presented in the source as a conjectural vanishing theorem; no resolution is supplied here.