Non-birationality conjecture for hyperplane restrictions of maps from very general hypersurfaces

Let $X_d\bsubseteq \mathbb{P}^{n+1}$ be a very general hypersurface of degree $d\geq n+3$, and let $Y$ be a smooth projective $n$-fold that is non-uniruled. Let $H_d:=X_d\cap H$, where $H$ is a very general hyperplane of $\mathbb{P}^{n+1}$. Suppose there is a dominant rational map $f:X_d\dashrightarrow Y$ with $\operatorname{deg} f>1$. Hyperplane-restriction conjecture. Then the restriction $f|_{H_d}$ cannot be birational to its image.

This conjecture is proposed as an inductive step toward the rational connectedness conjecture: restricting to a very general hyperplane should retain non-birationality when the target is non-uniruled. Its resolution status is not specified in the supplied text.