Contractibility of space of stability conditions on the projective plane via global dimension function

We compute the global dimension function $\mathrm{g}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{m}$ on the principal component ${\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}}^{\mathrm{†}}({\mathbb{P}}^2)$ of the space of Bridgeland stability conditions on ${\mathbb{P}}^2$. It admits $2$ as the minimum value and the preimage ${\mathrm{g}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{m}}^{-1}(2)$ is contained in the closure $\overline{{\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}}^{\mathrm{G}\mathrm{e}\mathrm{o}}({\mathbb{P}}^2)}$ of the subspace consisting of geometric stability conditions. We show that ${\mathrm{g}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{m}}^{-1}[2,x)$ contracts to ${\mathrm{g}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{m}}^{-1}(2)$ for any real number $x\ge 2$ and that ${\mathrm{g}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{m}}^{-1}(2)$ is contractible.