Determine the eventual connecting gonality of very general hypersurfaces

Determine whether the p-connecting gonality of a very general smooth hypersurface X of degree d in projective space of dimension n+1 equals d-1 for all sufficiently large p, under the condition d\geq 2n+2.

Background

The paper proves that for every smooth hypersurface X_d\subseteq \mathbb{P}^{n+1} of dimension n and every p\geq 2, its p-connecting gonality lies between d-n and d-1. The upper bound comes from projection from a point of X_d, which gives a rational covering of degree d-1.

For a very general hypersurface of degree d\geq 2n+2, the cited results suggest that the upper bound d-1 should be the exact eventual value as p becomes large. The authors explicitly state this as a likely but unverified assertion.