The unbendable-rational-curves conjecture for Fano manifolds

Let $X$ be a Fano manifold with Picard number one. A rational curve $f:\mathbb P^1\to X$ is unbendable when $f^*TX$ decomposes as

$$f^*TX\simeq\mathcal O(2)\oplus\mathcal O(1)^p\oplus\mathcal O^q$$

for some nonnegative integers $p,q$. The unbendable-rational-curves conjecture. There exists an irreducible family of unbendable rational curves on $X$ whose tangent vectors span the tangent spaces of $X$ at general points. Such a family would provide a geometric route toward simplicity of $TX$; the source establishes the corresponding existence for smooth horospherical varieties of Picard number one, but leaves the general Fano case as a question.