Vanishing conjecture for squares of ample generators on fake projective planes

Let $S$ be a fake projective plane, let $G$ be an ample generator of $\operatorname{Pic}(S)$, and let $\sigma$ be a torsion line bundle on $S$. Vanishing conjecture. One has

$$H^0(S,G^2\otimes\sigma)=0$$

for every torsion line bundle $\sigma$. This conjecture is motivated by the existence problem for exceptional collections in the derived category of coherent sheaves on a fake projective plane and is presented as a variant of conjectures attributed to Galkin, Katzarkov, Mellit and Shinder, and to Lai and Yeung. Its resolution is not indicated in the supplied source.