The 4D Barth conjecture for framed sheaf and ADHM moduli spaces

Let $(\vec{r},n)$ be the discrete parameters for the 4D ADHM quiver, and let $M_{Q_4}(\vec{r},n)$ be its stable representation moduli space. Let $M_{(\mathbb P^1)^4}(\vec{r},n)$ be the moduli space of framed sheaves on $(\mathbb P^1)^4$ with the corresponding parameters, equipped with their $\mathbb T$-equivariant 3-term symmetric obstruction theories. The 4D Barth conjecture. For any $(\vec{r},n)$, there exists a $\mathbb T$-equivariant isomorphism

$$M_{Q_4}(\vec{r},n) \cong M_{(\mathbb P^1)^4}(\vec{r},n)$$

intertwining the $\mathbb T$-equivariant 3-term symmetric obstruction theories on both sides. Both spaces have the same discrete parameters, $\mathbb T$-equivariant derived enhancements with a $(-2)$-shifted symplectic form, and the same virtual dimension. The conjecture is supported by the isomorphism of their $(\mathbb C^*)^r$-fixed loci and the coincidence of their virtual tangent representations at $\mathbb T$-fixed points, but the global isomorphism remains unproved.