Finite-strategy conjecture for normal quasicones

Let $\mathfrak{g}$ be an affine Kac–Moody algebra, let $V$ be a weight module, and let $\mathfrak{C}$ be the set of normal quasicones arising as annihilators of vectors in $V$. Let $\mathfrak{S}$ denote the set of strategies, where a strategy is a finite composition of root operators satisfying the stated nonvanishing, quasicone, and root-sum conditions. Finite-strategy conjecture. There is a finite set of strategies $\mathcal{S}\subset\mathfrak{S}$ such that the number of normal quasicones where no strategy succeeds is zero, namely

$$\bigcap_{C\in\mathfrak{C}}\left\{\#\operatorname{Ann}(sv)\geq\#C\text{ for all }s\in\mathcal{S}\right\}=\emptyset.$$

If true, this would ensure that every normal quasicone admits a successful strategy and support the proposed classification method. The supplied source gives no resolution status.