The genus-2 mutation invariance conjecture for tau invariants

Let $K\subset S^3$ be a knot, and let $\Sigma$ be a genus-2 surface in its complement. Genus-2 mutation is the operation of cutting $S^3$ open along $\Sigma$ and regluing the pieces by the hyperelliptic involution of $\Sigma$; the resulting knot is called a genus-2 mutant. Let $\tau^\sharp$ be the framed-instanton concordance homomorphism and let $\tau$ be the Heegaard Floer concordance invariant.

Genus-2 mutation invariance conjecture. The values of $\tau^\sharp$ and $\tau$ are preserved by genus-2 mutation.

The source notes that preservation of $\tau$ under Conway mutation is open in general, and that the corresponding question for arbitrary genus-2 mutation is less clear. The same issue applies to $\tau^\sharp$; no resolution is given here.