Classification of two-dimensional leaves in quiver Coulomb branches

Let $\mathbf{v}$ be a dimension vector, let $\GL_{\mathbf{v}}$ be the corresponding product of general linear groups, and let $\mathbf{N}_{\mathbf{v}}$ be the associated representation. Assume that $\mathcal{M}(\GL_{\mathbf{v}},\mathbf{N}_{\mathbf{v}})$ has no zero-dimensional leaves. Conditions (i) and (ii) are the two classes described in the preceding discussion: condition (i) is the modified goodness condition that every monopole operator not of the form $\pi_{\mathbf{v}}$ has degree at least $2$, while condition (ii) consists of the affine-support multiples of the primitive imaginary coroot and the Jordan-quiver cases with $v>1$. Two-dimensional-leaf classification conjecture. The Coulomb branch $\mathcal{M}(\GL_{\mathbf{v}},\mathbf{N}_{\mathbf{v}})$ has a $2$-dimensional leaf if and only if one of conditions (i) or (ii) is satisfied, and in each case there is a unique such leaf. In particular, if $\mathcal{M}(\GL_{\mathbf{v}},\mathbf{N}_{\mathbf{v}})$ is not weakly conical, then it has no two-dimensional leaves. The conjecture would yield the cited classification of isolated symplectic leaves; the source derives the result in the weakly conical setting but leaves the general case open.