Chow stability from tangent-bundle stability for smooth projective varieties

Let $X\subset\mathbb{P}^n$ be a complex irreducible smooth variety, and let $T\mathbb{P}^n_{|X}$ denote the restriction to $X$ of the tangent bundle of $\mathbb{P}^n$. Stability is taken with respect to $\mathcal{O}_X(1)$. Chow stability conjecture. If $T\mathbb{P}^n_{|X}$ is $\mathcal{O}_X(1)$-stable, then $X\subset\mathbb{P}^n$ is Chow stable. The paper proves the analogous implication for smooth curves and proposes this higher-dimensional extension; its resolution is not established in the supplied text.