The logarithmic Dolbeault equation conjecture for simple normal crossing divisors

Let $X$ be a compact Kähler manifold and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor in $X$. Let $L$ be a holomorphic bundle over $X$ such that

$$L^N=\mathcal{O}_X\left(\sum_{i=1}^r a_iD_i\right),\qquad 0\leq a_i\leq N,$$

where $a_i\in\mathbb{Z}$. The logarithmic Dolbeault equation conjecture. For any $\alpha\in A^{0,q}(X,\Omega^p_X(\log D)\otimes L^{-1})$ satisfying $\bar\partial\partial\alpha=0$, there exists

$$x\in A^{0,q-1}(X,\Omega^{p+1}_X(\log D)\otimes L^{-1})$$

such that

$$\bar\partial x=\partial\alpha.$$

Such a general logarithmic $\bar\partial$-equation would extend known $E_1$-degeneration and unobstructed-deformation results from smooth projective varieties to compact Kähler manifolds.