Weaker Borcherds conjecture on vanishing of Tate cohomology for Fricke elements

Let $V$ be a self-dual integral form of the monster vertex algebra, let $g$ be a Fricke element, and suppose that $g^j$ is a Fricke element for every integer $j>0$. Weaker Borcherds vanishing conjecture. Then

$$\hat{H}^1(g,V)=0.$$

This is proposed as a weaker version of Borcherds's conjecture on the vanishing of $hat{H}^1(g,V)$. The paper gives a counterexample to the stronger vanishing claim for all Fricke elements, while this restricted formulation is presented as a conjecture and its resolution is not supplied here.