Kovács's exterior-power characterization conjecture

Let $X$ be a smooth complex projective $n$-dimensional variety, let $\mathcal{F}$ be an ample vector bundle on $X$ that is a subsheaf of $\wedge^pT_X$, and let $p>0$. Assume that $\mathcal{F}=\wedge^p\mathcal{E}$ for some ample vector bundle $\mathcal{E}$. Kovács's exterior-power characterization conjecture. Then either

$$X\simeq \mathbb{P}^n,$$

or $p=n$ and

$$X\simeq Q_n.$$

This generalizes the Araujo–Druel–Kovács characterization theorem to higher-rank ample exterior-power subsheaves. The supplied context attributes the conjecture to S. Kovács and says that it was mentioned by K. Ross; no resolution is given.