The generalized volume bounds for Gorenstein Fano varieties with canonical singularities

Let $X$ be a Gorenstein Fano variety with canonical singularities. The source's Theorem A' gives, for $d=2$, $(-K_X)^2\leq 9$ with equality if and only if $X\cong\mathbb{P}^2$; for $d=3$, $(-K_X)^3\leq72$ with equality if and only if $X\cong\mathbb{P}(3,1,1,1)$ or $X\cong\mathbb{P}(6,4,1,1)$; and for $d\geq4$, $(-K_X)^d\leq2t_{d-1}^2$ with equality if and only if $X\cong\mathbb{P}(Q'_d)$. Generalized volume-bounds conjecture. The results of Theorem A' hold for Gorenstein Fano varieties with canonical singularities. The toric version is proved in the source; the extension beyond the toric case is posed as a conjecture.