$\mathbb Q$-Fano threefolds of index $7$

We show that if the inequality $\dim\mathopen|-K_X|\ge15$ holds for a $\mathbb Q$-Fano threefold $X$ of Fano index $7$, then $X$ is isomorphic to one of the following varieties: $\mathbb P(1^2,2,3)$, $X_6\subset\mathbb P(1,2^2,3,5)$, or $X_6\subset\mathbb P(1,2,3^2,4)$.