HOMFLYPT quantum one-cocycle evaluation conjecture

Let $K$ be a long knot, let $v_2(K)$ be its Vassiliev invariant of degree $2$, and let $P_K$ be its HOMFLYPT polynomial. Let $\delta$ denote the HOMFLYPT polynomial of the trivial $2$-component link, and let $\bar R^{(1)}$ be the combinatorial $1$-cocycle based on the HOMFLYPT invariant. Then

HOMFLYPT quantum one-cocycle conjecture.

$$\bar R^{(1)}(\operatorname{rot}(K))=-\delta v_2(K)P_K.$$

The conjecture is proved in the paper for the trivial knot, the trefoil, and the figure-eight knot; its general case remains open in the supplied text.