Logarithmic vanishing theorems for effective $q$-ample divisors

Let $X$ be a compact K\"ahler manifold and $D$ be a simple normal crossing divisor. If $D$ is the support of some effective $k$-ample divisor, we show $$H^q(X,\Omega^p_X(\log D))=0,\quad \text{for}\quad p+q>n+k.$$