Global-dimension contractibility conjecture for stability conditions on the projective plane

Let $\operatorname{D}^b(\operatorname{Coh}\mathbb{P}^2)$ be the bounded derived category of coherent sheaves on the projective plane, and let $\operatorname{Stab}\operatorname{D}^b(\operatorname{Coh}\mathbb{P}^2)$ carry the global-dimension function $\operatorname{gldim}$. A stability condition is geometric if all skyscraper sheaves $\mathcal{O}_x$ are stable with the same phase. Projective-plane global-dimension conjecture. The locus

$$\operatorname{gldim}^{-1}(2)$$

consists of almost all geometric stability conditions, and, for every real number $s>2$, the sublevel set

$$\operatorname{gldim}^{-1}[2,s)$$

contracts $\operatorname{gldim}^{-1}(2)$. The source says this was to be proved in forthcoming work, but supplies no resolution here.