The degree-of-irrationality conjecture for generic hypersurfaces

Let $X\subset \mathbb{P}^n$ be a generic hypersurface of degree $d\geq 2n-1$, and let $\mathrm{d}_r(X)$ denote its degree of irrationality. A dominant rational map is a rational map with dense image. Degree-of-irrationality conjecture. The degree of irrationality of $X$ is

$$\mathrm{d}_r(X)=d-1.$$

If $d\geq 2n$, any dominant rational map

$$X\dashrightarrow \mathbb{P}^{n}$$

of degree $d-1$ is obtained by projecting $X$ from one of its points. The claim would extend Noether's theorem on the degree of irrationality of hypersurfaces; the stated context does not establish whether it has been resolved.