Conjecture on special manifolds and lifted entire curves

Let $X$ be a projective manifold, let $X_1:=\mathbb P(T_X)$ be the projectivized tangent bundle, and for every entire curve $f:\mathbb C\to X$ let $f_{[1]}:\mathbb C\to X_1$ be its lift. Define

$$\operatorname{Exc}_1(X)\subset X_1$$

to be the Zariski closure of the union of the images $f_{[1]}(\mathbb C)$ of all lifted entire curves. Conjecture on lifted entire curves. $X$ is not special if and only if

$$\operatorname{Exc}_1(X)\neq X_1.$$

This conjecture aims to characterize non-special projective manifolds through the exceptional locus of lifted entire curves; the source does not state what cases are known or whether it remains open.