Wall-crossing conjecture for perverse coherent systems on the local resolved conifold

Let $X=\mathcal{O}_{\mathbb{P}^1}(-1,-1,0)$, let $T_0$ be the Calabi–Yau torus acting on $X$, and let $\Theta$ lie on one of the walls $L_{\pm}^{-}(k)$ or $L_{\pm}^{+}(k)$. For a $T_0$-fixed $\Theta$-stable perverse coherent system $I_0$, let $S_{k-1}$ be the wall-supported $T_0$-fixed $\Theta$-stable perverse coherent sheaf and set

$$P_{k-1,r}^{I_0}=\left\{I_0\oplus S_{k-1}^{\oplus r}[-1]\right\},\qquad r\geqslant0.$$

Write $\Theta_{\pm}=(\theta_0\mp0^+,\theta_1)$, let $\pi_+$ and $\pi_-$ be the maps from the adjacent chambers to the wall, and let $\lambda_3$ be as above.

Wall-crossing conjecture. If $\Theta\in L_-^-(k)$ or $L_+^-(k)$ with $k\geqslant1$, then there exist choices of signs such that

$$\frac{\sum_r t^r\sum_{I\in\pi_+^{-1}(P_{k-1,r}^{I_0})}e_{T_0}(\chi_X(I,I)^{\frac12}_0)\cdot e_{T_0\times\mathbb{C}^*}(\chi_X(F)^\vee\otimes e^m)}{\sum_r t^r\sum_{I\in\pi_-^{-1}(P_{k-1,r}^{I_0})}e_{T_0}(\chi_X(I,I)^{\frac12}_0)\cdot e_{T_0\times\mathbb{C}^*}(\chi_X(F)^\vee\otimes e^m)}=(1-t)^{k\frac{m}{\lambda_3}}.$$

If $\Theta\in L_-^+(k)$ or $L_+^+(k)$ with $k\geqslant0$, then there exist choices of signs such that the same ratio equals

$$(1-t^{-1})^{k\frac{m}{\lambda_3}}.$$

This conjecture predicts that the wall-crossing factor is independent of the initial stable object $I_0$. It is motivated by the analogous wall-crossing formula of Nagao and Nakajima for the resolved conifold and is intended as its fourfold analogue.