RC-positivity, ampleness, and positivity of tautological bundles

Let $X$ be a projective manifold and let $\mathscr E$ be a holomorphic vector bundle on $X$. Write ${\mathcal O}_{\mathscr E}(1)$ for the tautological line bundle on ${\mathbb P}(\mathscr E^*)$, and let $\dim X$ denote the dimension of $X$.

RC-positivity equivalence conjecture. The following statements are equivalent:

1. ${\mathcal O}_{\mathscr E^*}(-1)$ is RC-positive.
2. ${\mathcal O}_{\mathscr E}(1)$ is $(\dim X-1)$-ample.
3. ${\mathcal O}_{\mathscr E}(1)$ is $(\dim X-1)$-positive.
4. $\mathscr E$ is RC-positive.

The source says that $(4)\Longrightarrow(3)\Longrightarrow(2)\Longleftrightarrow(1)$ are known, and that the equivalence is proved when $\operatorname{rank}(\mathscr E)=1$ or $\dim X=1$. The remaining implications are open in general.