The genus-2 mutation invariance conjecture for Floer homology ranks

Let $Y$ be a closed $3$-manifold and let $\Sigma\subset Y$ be a separating genus-2 surface. Its genus-2 mutant $Y'$ is obtained by cutting $Y$ open along $\Sigma$ and regluing by the hyperelliptic involution. Let $I^\#$ denote framed instanton homology and let $\widehat{\mathit{HF}}$ denote hat Heegaard Floer homology.

Floer-rank mutation conjecture. The ranks of $I^\#$ and $\widehat{\mathit{HF}}$ are preserved by genus-2 mutation.

Genus-2 mutation preserves classical homology but need not preserve $\operatorname{Spin}^c$-graded Heegaard Floer homology. The source reports evidence for preservation of the total Heegaard Floer rank and conjectures the analogous statement for both Heegaard Floer and framed instanton homology.