Degeneration at the third page for plane-curve spectral sequences

Let $C:f=0$ be any reduced plane curve, and let $E_r^{s,t}(f)_k$ denote the terms of the spectral sequences associated with $C$.

Degeneration conjecture. The spectral sequences degenerate at the $E_3$-term for any reduced plane curve $C:f=0$.

This conjecture is known when all singularities of $C$ are weighted homogeneous, by the stated corollary, but the authors report that they do not know how to prove it or even check it for a specific example in general.