Folklore openness conjecture for pseudo-Brody hyperbolicity

Let $f:\mathscr{X}\to\mathbb{D}$ be a holomorphic proper submersion from a complex manifold to the unit disk with connected fibers, and write $X_t=f^{-1}(t)$. A fiber is pseudo-Brody hyperbolic if every nonconstant holomorphic map from $\mathbb C$ to it has image in a proper Zariski closed subset. Folklore openness conjecture. If $X_0$ is pseudo-Brody hyperbolic, then $X_t$ is also pseudo-Brody hyperbolic for sufficiently small $t$. The ordinary Brody hyperbolicity analogue is known, while the pseudo-Brody statement is presented as folklore and no resolution is given.