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

We compute the global dimension function $\mathrm{gldim}$ on principal component $\mathrm{Stab}^{\dag}(\mathbb{P}^2)$ of space Bridgeland stability conditions $\mathbb{P}^2$. It admits $2$ as minimum value and preimage $\mathrm{gldim}^{-1}(2)$ is contained in closure $\bar{\mathrm{Stab}^{\mathrm{Geo}}(\mathbb{P}^2)}$ subspace consisting geometric conditions. show that $\mathrm{gldim}^{-1}[2,x)$ contracts to for any real number $x\geq 2$ contractible.