Anticanonical Volumes of Terminal and Canonical Gorenstein Fano Fourfolds

Let $X$ be a complex $\mathbb{Q}$-factorial Gorenstein Fano fourfold of Picard number one. We prove that, if $X$ has terminal singularities, then $(-K_X)^4\le648$, with equality precisely for $\mathbb{P}(1,1,1,1,2)$. If the singularities are canonical, we obtain $(-K_X)^4\le7332$. Under the addition