Manolescu's graded HOMFLY-PT to knot Floer homology spectral sequence conjecture

Let $L$ be a link in $S^3$. The HOMFLY-PT homologies carry gradings $\operatorname{gr}_T$ and $\operatorname{gr}_M$, and the grading-shifted knot Floer homologies $coverline{\operatorname{HFK}}'(L)$, $(\operatorname{HFK}^-)'(L)$, and $\operatorname{HFK}'(L)$ are defined using the corresponding grading shifts. Manolescu's graded conjecture. There are spectral sequences whose pages are bigraded by $(\operatorname{gr}_T,\operatorname{gr}_M)$, whose differential on each page has degree $(0,1)$, and whose pages are the bigraded homology of the preceding page, with

• $E_2$ page $coverline{H}(L)$ and $E_{\infty}$ page $coverline{\operatorname{HFK}}'(L)$;
• $E_2$ page $H^-(L)$ and $E_{\infty}$ page $(\operatorname{HFK}^-)'(L)$;
• $E_2$ page $H(L)$ and $E_{\infty}$ page $\operatorname{HFK}'(L)$,

as bigraded vector spaces. This is the grading-refined form of the proposed HOMFLY-PT to knot Floer spectral sequences, and it specifies the grading behavior needed for comparisons between the theories. Identifying the relevant $E_2$ pages with HOMFLY-PT homology remains open in the link setting.