The light-factor conjecture for weight-maximizing differential summands

Let $M_{n}^{J}$ be the degree-$n$ piece at filtration level $J$, let $\mathrm{wt}(\omega)$ denote the weight of $\omega$, and let $d_{\sim}$ be the indicated component of the differential. A factor in a summand is light when it has the lower weight type defined in the surrounding construction. Light-factor conjecture. Suppose the weight of $\omega\in M_{n}^{J}$ is maximized only by summands of $d_{\sim}\omega$ (for example, when $J=1$). Then some weight-maximizing summand of $d\omega$ has a light factor.

The claim is intended to extend the degree-two weight argument to higher degrees. The preceding corollary establishes the relevant estimate in degree two, but the source says that extending the process to higher degrees is not clear; this conjecture is therefore open.