Let $X$ be a smooth projective variety and $D$ a log-canonical divisor and let $U = X \setminus D$. I have heard the slogan "log-canonical is just as good as normal crossings for Hodge theory". This question concerns how to make that slogan precise as applied to the cohomology of $U$.

If $D$ were normal crossings, then $H^{\ast}(U, \mathbb{C})$ would be the hypercohomology of the complex $\Omega^p(\log D)$ on $X$. Moreover, the spectral sequence of that hypercohomology would degenerate at $E_1$. I could compute the weight filtration on $H^{\ast}(U, \mathbb{C})$ by filtering $\Omega^p(\log D)$ according to how many poles my $p$-forms have along $D$.

What are the analogous statements for a log-canonical divisor? Let me emphasize that I would like statements which are true on $X$, without passing to a resolution of singularities.