Dowlin's sl(n) to knot Floer homology spectral sequence conjecture

Let $L$ be a link in $S^3$ and let $n$ be a positive integer. Write $\overline{H}_n(L)$ and $H_n(L)$ for reduced and unreduced $\mathfrak{sl}(n)$ homology, and $coverline{\operatorname{HFK}}'_n(L)$ and $\operatorname{HFK}'_n(L)$ for the corresponding grading-shifted knot Floer theories. The theories are graded by $\frac{1}{n}\mathbb{Z}$. Dowlin's conjecture. There exist spectral sequences whose pages are graded by $\frac{1}{n}\mathbb{Z}$, whose differentials have degree $+1$, and whose pages are the $\frac{1}{n}\mathbb{Z}$-graded homology of the preceding page, with

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

as $\frac{1}{n}\mathbb{Z}$-graded vector spaces. These spectral sequences would relate reduced and unreduced $\mathfrak{sl}(n)$ homology to the corresponding knot Floer theories for links. The conjecture is stated as a grading-refined version of Dowlin's proposed spectral sequences; no resolution is supplied here.