Polynomial-time conjecture for Khovanov homology of fixed-strand closed braids

Let $b$ be a braid with a fixed number of strands, let $\widehat{b}$ denote its closure, and let the number of crossings of $b$ be the input size. Przytycki–Silvero conjecture. Computing the Khovanov homology $\operatorname{Kh}(\widehat{b})$ of a closed braid with fixed number of strands has polynomial time complexity with respect to the number of crossings. The conjecture concerns whether the tractability known for the Jones polynomial of fixed-strand braid closures extends to Khovanov homology. The paper proves this in the case of $3$-braids and studies related algorithms for more general braids, while the general fixed-strand claim remains open in the supplied context.