Extension of the Chern-character filtration theorem to arbitrary-rank bundles

Let $M$ be the polarized manifold under consideration, let $S_V^q\operatorname{Vect}(M)$ denote the corresponding filtration of vector bundles, and let $S_H^{p-q}$ and $\bar S_H^{p-q}$ be the Hodge filtrations on $H^{2p}(M,\mathbb{Q})$. For a vector bundle $E$ of arbitrary rank, write $\operatorname{ch}_p(E)$ for the degree-$2p$ component of its Chern character.

Chern-character filtration conjecture.

$$\operatorname{ch}_p(S_V^q\operatorname{Vect}(M))\subset (S_H^{p-q}\cap \bar S_H^{p-q})H^{2p}(M,\mathbb{Q}).$$

The preceding theorem establishes this inclusion for line bundles. The conjecture asserts that the same compatibility between the vector-bundle filtration and the Hodge filtration holds in arbitrary rank; no resolution is given here.