Floer-theoretic fibred Dehn twist exact sequence conjecture

Suppose that $(M,\omega)$ and $(\bar{M},\bar{\omega})$ are symplectic, that $\rho\colon V\to\bar{M}$ is an $S^k$-bundle with structure group $\operatorname{SO}(k+1)$, and that $i\colon V\hookrightarrow M$ is an embedding with $i^*\omega=\rho^*\bar{\omega}$. Let $\phi\in\operatorname{Aut}(M,\omega)$ preserve $V$ and cover $\bar{\phi}\in\operatorname{Aut}(\bar{M},\bar{\omega})$. Let $e=e(V)\cap\cdot$ be the Euler-class endomorphism of $HF_*(\bar{\phi})$, and let $\operatorname{cone}(e)$ be its mapping cone. Under the monotonicity and minimal-Maslov-index hypotheses ensuring that the maps $a$ and $b$ are defined, there are maps $a\colon H_*\operatorname{cone}(e)\to HF_*(\phi)$ and $b\colon HF_*(\phi)\to HF_*(\phi\circ\tau_V)$.

Fibred Dehn twist exact sequence conjecture. Under conditions which render $a$ and $b$ well-defined, they fit into a long exact sequence

$$H_*\operatorname{cone}(e)\xrightarrow{a}HF_*(\phi)\xrightarrow{b}HF_*(\phi\circ\tau_V)\longrightarrow H_{*-1}\operatorname{cone}(e).$$

This is presented as a speculative consequence for Floer homology and is the subject of work in progress. The stated construction gives the maps under suitable monotonicity and minimal Maslov index assumptions, but the exactness assertion remains open.