Stable/co-stable wall-crossing conjecture for 4D ADHM invariants

Let $M_{Q_4}(\vec{r},n)$ be the moduli space defined using the stable condition, and let $M_{Q_4}^c(\vec{r},n)$ be the moduli space defined using co-stability, namely the condition that there is no nonzero subspace $V'\subseteq\bigcap_A\ker(J_A)$ invariant under $B_1,\dots,B_4$. Let $\widehat{\mathcal O}^{\mathrm{vir}}$ denote the virtual structure sheaf on either moduli space. Stable/co-stable wall-crossing conjecture. For all $\vec{r}$ and $n$,

$$\chi(M_{Q_4}(\vec{r},n),\widehat{\mathcal O}^{\mathrm{vir}})=\chi(M_{Q_4}^c(\vec{r},n),\widehat{\mathcal O}^{\mathrm{vir}}).$$

Equivalently, the wall-crossing between the corresponding invariants is conjectured to be trivial. The claim is proved in a special two-dimensional reduction and has been verified in several low-rank and low-instanton cases, but remains open in general.