Baldwin–Levine's mutation-invariance conjecture for delta-graded knot Floer homology

Let $L$ and $L'$ be a mutant pair of links. Write ${\rm {\widehat{HFK}}}_{\delta}(L)$ for the knot Floer homology obtained by collapsing the bigrading along diagonals $m-a=\delta$. Baldwin–Levine's conjecture. There is an isomorphism

$${\rm {\widehat{HFK}}}_{\delta}(L)\cong {\rm {\widehat{HFK}}}_{\delta}(L').$$

Computations for knots with 11 and 12 crossings support this claim, while the paper develops a strategy and proves it for a large class of tangles; the general case remains open.