The pretzel-minimizer conjecture for 2-almost positive diagrams

A knot is 2-almost positive if the minimal number of negative crossings among all its diagrams is $2$. Let $L$ be a 2-almost positive diagram with even crossing number that minimizes the third Vassiliev invariant $v_3$ among all diagrams with that crossing number. The pretzel-minimizer conjecture. Then $L$ is a diagram of a pretzel knot of type $(a_1,\dots,a_k,2)$, where

$$a_i\in\{1,3\}.$$

The conjecture is motivated heuristically by examples involving 2-almost positive knots, while the text explains that the preceding approach does not classify such diagrams; no resolution is supplied.