Degeneration and log mixed Hodge structure conjecture for logarithmic de Rham cohomology

Let $f:X\to S$ satisfy condition D1.1, and let

$$H^m(X/S):=(H^m(X/S)_{\mathbb{Z}},H^m(X/S)_{\cO},W,F)$$

be the associated $4$-ple, with Hodge filtration $F$ and weight filtration $W$ as defined above. The Hodge-to-de Rham spectral sequence is

$$E^{p,q}_1=R^qf_*(\omega^p_{X/S})\Rightarrow H^{p+q}_{\operatorname{dR}}(X/S).$$

Degeneration and LMH conjecture. (1) This spectral sequence degenerates at $E_1$, and all $R^qf_*(\omega^p_{X/S})$ are locally free of finite rank as $\cO_S$-modules; hence $H^m(X/S)$ is a pre-log mixed Hodge structure on $S$. (2) The pre-log mixed Hodge structure $H^m(X/S)$ is a log mixed Hodge structure on $S$. This is presented as one of the main problems in log Hodge theory; the supplied source does not establish either assertion, so their resolution remains open.