Griffiths' conjecture on positivity of Chern–Weil forms

Let $(E,h)\to X$ be a Griffiths semipositive Hermitian holomorphic vector bundle. Let $P$ be a non-negative combination of Schur polynomials, and let $P\big(c_{\bullet}(E,h)\big)$ denote the differential form obtained by formally evaluating $P$ in the Chern forms of $(E,h)$. Griffiths' conjecture. The differential form

$$P\big(c_{\bullet}(E,h)\big)$$

is positive. This is a classical positivity question for characteristic forms associated with Griffiths semipositive vector bundles; the source describes the statement as a modification of Griffiths' original conjecture.