Contractibility conjecture for the principal stability component of $\mathbb P^3$

Let $\mathrm{Stab}(\mathbb P^3)$ be the space of Bridgeland stability conditions on $\mathbb P^3$, and let $\mathrm{Stab}^{\dagger}(\mathbb P^3)\subset\mathrm{Stab}(\mathbb P^3)$ be the principal connected component containing the image of $\Sigma_{\Psi}$. Let $\mathrm{Stab}^{\mathrm{Geo}}(\mathbb P^3)$ and $\mathrm{Stab}^{\mathrm{Alg}}(\mathbb P^3)$ denote the geometric and algebraic stability-condition subspaces, respectively.

Contractibility conjecture. The principal connected component is the union of the geometric and algebraic stability conditions and is contractible:

$$\mathrm{Stab}^{\dagger}(\mathbb P^3)=\mathrm{Stab}^{\mathrm{Geo}}(\mathbb P^3)\bigcup\mathrm{Stab}^{\mathrm{Alg}}(\mathbb P^3).$$

The conjecture concerns the homotopy type of a principal component of the stability-condition space; contractibility would describe how the geometric and algebraic loci together account for this component. The source presents it as an open conjecture inspired by prior work on stability-condition spaces.