Alternating knot with unknotting number two among the 959 unresolved cases

Determine whether at least one of the 959 prime alternating knots with retained unknotting-number range [2,3], for which no crossing change in a minimal diagram yields a knot of unknotting number one, has unknotting number two.

Background

For 959 alternating knots, the authors verify that no crossing change in any minimal diagram produces a knot of unknotting number one. Consequently, each such knot has strong Bernhard–Jablan unknotting number at least three; if any of them has unknotting number two, it would be an alternating counterexample to the strong Bernhard–Jablan conjecture. Their recorded ranges remain [2,3], so the question is unresolved.