Andreatta–Wiśniewski projective-space characterization conjecture

Let $X$ be a smooth projective variety, let $\mathscr{E}$ be an ample vector bundle on $X$, and let $T_X$ denote the tangent bundle of $X$. Andreatta–Wiśniewski's conjecture. If

$$\operatorname{Hom}(\mathscr{E},T_X)\neq 0,$$

then $X\simeq\mathbb{P}^n$. This conjecture is presented as a plausible strengthening of a result of Andreatta and Wiśniewski and is used to reduce the general case of Sommese's classification conjecture.