The covering gonality conjecture for very general hypersurfaces

Let $X\subset \mathbb{P}^{n+1}$ be a very general hypersurface of degree $d\geqslant 2n$. Here $\operatorname{cov.gon}(X)$ denotes the covering gonality of $X$, namely the least gonality of a curve in a family of curves covering $X$. Covering gonality conjecture.

$$\operatorname{cov.gon}(X)=d-\left\lfloor\frac{\sqrt{16n+1}-1}{2}\right\rfloor.$$

The preceding discussion explains that plane curves with a singularity of the indicated multiplicity give the corresponding upper bound. The equality is proposed in the context of determining whether such plane curves compute the covering gonality, but the source does not establish it for all the stated values of $n$ and $d$.