Hochschild cohomology quotient conjecture for equivariant partially wrapped Fukaya categories

Let $X$ be a Liouville sector equipped with a Hamiltonian $\bS^1$-action, and suppose that its symplectic quotient $\underline{X}$ at the zero moment level is smooth and is itself a Liouville sector. Let $\mathcal{W}(X)$ denote the wrapped Fukaya category of $X$, and consider the partially wrapped Fukaya category of $\bS^1$-invariant objects in $X$.

Hochschild cohomology quotient conjecture. There exists a $\mathbb{C}[s]$-module structure on $HH^*(\mathcal{W}(X))$ such that $HH^*(\mathcal{W}(\underline{X}))$ is isomorphic to the quotient of the Hochschild cohomology of the partially wrapped Fukaya category of $\bS^1$-invariant objects in $X$ by the ideal generated by $(s-1)$.

This is presented as a homological-mirror-symmetry reformulation of the preceding mirror fibration conjecture in the case $G=\bS^1$. The supplied text gives no resolution status or further theorem establishing the asserted isomorphism.