The alpha-invariant conjecture for hypersurfaces without star points

Let $X_d\subset\mathbb{P}^{n}$ be the hypersurface of degree $d$ considered above, let $H_{X_d}$ denote its hyperplane class, and let $\alpha(X_d,H_{X_d})$ be its alpha invariant. A star point is a point of $X_d$ for which the tangent hyperplane section is a cone with vertex at that point.

Alpha-invariant conjecture. If $X_d$ does not contain star points, then

$$\alpha(X_d,H_{X_d})>\frac{n}{d}.$$

The preceding results establish the lower bound $\alpha(X_d,H_{X_d})\geq\frac{n}{d}$ and show that equality holds when $X_d$ contains a star point. For plane curves, the claim follows from the cited corollary; the statement is presented as the natural expectation in the general setting.