The positive-diagram crossing-number conjecture

Let $D$ be a positive reduced diagram of a positive knot $K$, with crossing number $c(D)$, and let $g(K)$ denote the genus of $K$. The crossing number $c(K)$ is the minimum crossing number among diagrams of $K$.

Positive-diagram crossing-number conjecture.

$$c(K)\ge c(D)-2g(K)+1.$$

This would substantially improve the known bound $c(K)\ge\sqrt{2c(D)}$ for positive reduced diagrams. The source presents the estimate as appearing to be true, but supplies no resolution.