Positive tangent-bundle Seshadri constants characterize projective space

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field, and let $x_0\in X$. Write $\operatorname{\varepsilon}(TX;x_0)$ for the Seshadri constant of the tangent bundle at $x_0$. Positive tangent-bundle Seshadri constant conjecture. If there exists $x_0\in X$ such that

$$\operatorname{\varepsilon}(TX;x_0)>0,$$

then

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

This is proposed as an extension of the result proved for Fano varieties and for general points in characteristic zero; the unrestricted statement remains open.