The completeness conjecture for Jones rational coincidences

A rational knot is represented by an integer sequence presentation, and a Jones rational coincidence is a pair of rational knots with the same Jones polynomial. The pivoting-pairs template allows the move

$$({\bf n}_0^{\epsilon_0},d_1,{\bf n}_1^{\epsilon_1},\ldots,d_k,{\bf n}_k^{\epsilon_k})\longrightarrow({\bf n}_k^{\epsilon_k},d_k,{\bf n}_{k-1}^{\epsilon_{k-1}},\ldots,d_1,{\bf n}_1^{\epsilon_0})$$

for $d_1,\ldots,d_k\in\mathbb{Z}$, $\epsilon_0,\ldots,\epsilon_k\in\{1,*\}$, and integer sequences ${\bf n}_0\sim{\bf n}_1\sim\ldots\sim{\bf n}_k$; the second template is the move specified in Theorem 2 of the source. Completeness conjecture. All Jones rational coincidences can be obtained by using the moves in the templates of Theorems 3 and 2. This would classify all rational-knot pairs sharing a Jones polynomial in terms of the two templates, extending the constructive results supplied by those templates; whether every such coincidence is captured remains open.