Bounding the volumes of singular Fano threefolds

Let $(X,\mathrm{\Delta })$ be an $n$-dimensional $ϵ$-klt log $\text{QQ}$-Fano pair. We give an upper bound for the volume $\mathrm{V}\mathrm{o}\mathrm{l}(-(K_X+\mathrm{\Delta }))=(-(K_X+\mathrm{\Delta }){)}^n$ when $n=2$ or $n=3$ and $X$ is {$\text{QQ}$-factorial} of $\rho (X)=1$. This bound is essentially sharp for $n=2$. Existence of an upper bound for anticanonical volumes is related the Borisov-Alexeev-Borisov Conjecture which asserts boundedness of the set of $ϵ$-klt log $\text{QQ}$-Fano varieties of a given dimension $n$.