Conjecture on the space of Alexandrov embedded Lawson symmetric CMC surfaces

A Lawson symmetric CMC surface is a compact constant-mean-curvature surface in $\mathbb{S}^3$ of genus $g\geq 1$ with a cyclic symmetry of order $g+1$ having four fixed points. For each pair of integers $g\geq 1$ and $n\geq 1$, let $\Xi_g^n$ denote the known Lawson symmetric families, which also have the symmetry induced by the hyperelliptic involution, and let $\widehat\Xi_g^n$ denote the conjectured additional one-parameter families of Alexandrov embedded Lawson symmetric surfaces lacking that symmetry. Lawson symmetric surface space conjecture. The space of Alexandrov embedded Lawson symmetric constant-mean-curvature surfaces consists of the families $\Xi_g^n$ and $\widehat\Xi_g^n$. The conjecture asserts that the generalized Whitham flow from the $(2n+1)$-lobed Delaunay tori reaches the families $\widehat\Xi_g^n$, which converge to a chain of $(g+1)n+1$ constant-mean-curvature spheres. The completeness of this description would classify the Alexandrov embedded Lawson symmetric CMC surfaces beyond the currently known families; the supplied text gives no resolution, so the conjecture remains open.