Kohn-type conjecture for 2-adjacent alternating knots

Let $K$ be an alternating non-trivial $2$-adjacent knot, and let a minimal diagram mean a diagram of $K$ with the minimum crossing number. A $2$-adjacency set is a pair of crossings in a diagram that exhibits $2$-adjacency. Kohn-type conjecture. Every alternating non-trivial $2$-adjacent knot has a $2$-adjacency set in every minimal diagram. This is presented as a consequence of the two preceding conjectures: one asserting that the relevant unknotting crossings remain $2$-adjacent, and the other excluding same-sign $2$-adjacency except for the trefoil.