The stability-condition and quantum-cohomology correspondence for local $\mathbb P^2$

Let $X=\mathcal O_{\mathbb P^2}(-3)$, and let $\operatorname{Stab}^{0}_{n}(X)$ be the normalized connected submanifold of stability conditions defined by $Z(\mathcal O_x)=i$. Let $\mathbb A^2=\{(z_0,z_1,z_2)\in\mathbb C^3:z_0+z_1+z_2=i\}$, and let $\mathcal Z\colon\operatorname{Stab}^{0}_{n}(X)\longrightarrow\mathbb A^2$ send a stability condition to $(Z(S_0),Z(S_1),Z(S_2))$. Let $\tilde M^0/\mathbb C$ be the quotient of the relevant universal-covering space on the quantum-cohomology side, and let $\mathcal W\colon\tilde M^0/\mathbb C\longrightarrow\mathbb A^2$ be the homogeneous twisted period map. Stability-condition/quantum-cohomology conjecture. There is a commuting diagram

$$\begin{CD} \operatorname{Stab}^{0}_{n}(X) &@>F>> &\tilde M^0/\mathbb C\\ @V{\mathcal Z}VV && @V{\mathcal W}VV \\ \mathbb A^2 &@= &\mathbb A^2 \end{CD}$$

Moreover, $F$ is an isomorphism onto a dense open subset. This conjecture predicts that the normalized stability-condition space for the local Calabi–Yau threefold $X$ is related to the quantum-cohomology side by the homogeneous twisted period map. The source gives no evidence of a resolution, so the conjecture remains open.