Fibration property for algebraically hyperbolic varieties

Let $k$ be an algebraically closed field of characteristic zero. Let $f:X\to Y$ be a morphism of projective varieties over $k$. For $y\in Y(k)$, write $X_y$ for the fiber over $y$.

Fibration property. If $Y$ is algebraically hyperbolic over $k$ and, for all $y\in Y(k)$, the projective scheme $X_y$ is algebraically hyperbolic over $k$, then $X$ is algebraically hyperbolic over $k$.

The paper states that this conjecture follows from Demailly's conjecture, and notes that the analogous result for families of Kobayashi hyperbolic varieties is known. Thus it is open insofar as Demailly's conjecture remains open.