The generalized Eckardt point characterization for smooth hypersurfaces

Let $X$ be a smooth hypersurface of degree $n\geq 3$ in $\mathbb{P}^n$. A wild tiger for $X$ is an anticanonical divisor $S\in |-K_X|$ attaining the total log canonical threshold of $X$, and $LCS(X,cS)$ denotes the locus of log canonical singularities of the pair $(X,cS)$. An Eckardt point is a point $p\in X$ for which an anticanonical divisor is a cone in $\mathbb{P}^{n-1}$ over a smooth hypersurface of degree $n$ in $\mathbb{P}^{n-2}$ with vertex $p$.

Generalized Eckardt point conjecture. If the total log canonical threshold of $X$ is $\frac{n-1}{n}$, then a wild tiger $S$ for $X$ is a cone in $\mathbb{P}^{n-1}$ over a smooth hypersurface of degree $n$ in $\mathbb{P}^{n-2}$ with vertex $p$. Moreover,

$$LCS\left(X,\frac{n-1}{n}S\right)=\{p\},$$

where $p$ is an Eckardt point of $X$.

This conjectures that attaining the threshold $\frac{n-1}{n}$ characterizes the generalized Eckardt-point configuration; the preceding discussion establishes the analogous implication for smooth cubic surfaces and the theorem gives the forward construction from an Eckardt point. The general converse is left open.