General Floer homology and asymptotic growth conjecture for mapping classes

Let $M$ be a compact connected surface and let $\Gamma=\pi_0(\operatorname{Diff}^+(M))$. For any mapping class $g\in\Gamma$, let $\phi\in\operatorname{Symp}^m(M,\omega)$ be a monotone representative for a $\phi$-invariant area form $\omega$. Let $U$ be the neighborhood of the reducing curves, and let $M_{\mathrm{id}}$ be the union of the components of $M\setminus\operatorname{int}(U)$ on which $\phi$ restricts to the identity. If $\psi$ is a standard Thurston canonical-form representative of $g$, let $\lambda$ be the largest stretching factor of its pseudo-Anosov pieces, with $\lambda=1$ when there is no pseudo-Anosov piece. Then

General Floer homology and asymptotic growth conjecture.

$$HF_*(\phi)=H_*(M_{\mathrm{id}},\partial M_{\mathrm{id}};\mathbb{Z}_2)\oplus\mathbb{Z}_2^{N(\phi|_{M\setminus M_{\mathrm{id}}})},$$

and

$$F^{\infty}(g):=\operatorname{Growth}(\dim HF_*(\phi^n))=\lambda=h(\psi)=\limsup_{n\rightarrow\infty}|N(\psi^n)|^{1/n}.$$

This proposes a complete description of Floer homology for mapping classes in terms of the identity pieces and Nielsen data, together with an asymptotic growth formula governed by the largest pseudo-Anosov stretching factor. The supplied text does not state whether the conjecture has been resolved.