Tensor-product fixed-point conjecture for Coulomb branches

Let $W=W^1\oplus W^2$ be a decomposition inducing dominant coweights $\lambda^1$ and $\lambda^2$, let $\mathcal M^\nu(\lambda,\mu)$ be the corresponding deformation of the Coulomb branch, and let $\chi$ act on it as above. The tensor-product fixed-point conjecture. The fixed-point set $\mathcal M^\nu(\lambda,\mu)^\chi$ is finite and has a natural bijection with

$$\bigsqcup_{\mu=\mu^1+\mu^2}\mathcal M(\lambda^1,\mu^1)^\chi\times\mathcal M(\lambda^2,\mu^2)^\chi.$$

Together with the preceding fixed-point and geometric Satake conjectures, this would identify the nonempty fixed-point condition with the condition that $\mu$ is a weight of $V(\lambda^1)\otimes V(\lambda^2)$; it is not established in general.