Two-bridge knot transvergent diagram conjecture

Let $K$ be a two-bridge knot, written in Conway notation as $C(a_1,\ldots,a_n)$. A transvergent diagram is a diagram with the relevant transverse symmetry, and let $c(K)$ denote the crossing number of $K$. The conditions are: either $n$ is even, all $a_i>0$, and $a_2,a_4,\ldots$ are even; or $n$ is odd, all $a_i>0$, and the twist-numbers are symmetric, so that $a_1=a_n$, $a_2=a_{n-1}$, and so on. In the latter case, $a_{\frac{n+1}{2}}$ is necessarily odd.

Two-bridge knot transvergent diagram conjecture. A two-bridge knot $K$ has a transvergent diagram with $c(K)$ crossings if and only if one of these two conditions is fulfilled.

This conjecture characterizes when the minimal crossing number can be realized by a transvergent diagram. It is illustrated for two-bridge knots with crossing number less than $8$; the source provides no resolution of the general claim.