The periodic Floer homology conjecture for fibred knots

Let $K\subset Y$ be a fibered knot of genus $g$, with monodromy $\varphi$, and let $T_\varphi$ denote its mapping torus. Write $\overline{Y}$ and $\overline{K}$ for the orientation reversals of $Y$ and $K$, respectively, and let $\widehat{HFK}(\overline{Y},\overline{K},i)$ be the hat version of knot Floer homology in Alexander grading $i$. The group $PFH_{i+g}^\sharp(T_\varphi)$ is the sharp-version of periodic Floer homology in degree $i+g$. The periodic Floer homology conjecture. If $K\subset Y$ is a fibered knot with monodromy $\varphi$, then

$$\widehat{HFK}(\overline{Y},\overline{K},i)\cong PFH_{i+g}^\sharp(T_\varphi).$$

This is proposed by comparing sharp periodic Floer homology with sharp embedded contact homology and sutured contact homology. The source does not provide a resolution, so the conjecture remains open.