The Murasugi-Przytycki braid-index conjecture for alternating diagrams

Let $D$ be an alternating diagram of a link $L$. Write $b(L)$ for the braid index, $s(D)$ for the number of Seifert circles, and $\operatorname{ind}(D)$ for the index of the Seifert graph associated with $D$. Define

$$\operatorname{mpb}(D)=s(D)-\operatorname{ind}(D).$$

Murasugi-Przytycki conjecture. One has

$$b(L)=\operatorname{mpb}(D).$$

This conjecture concerns the sharpness of the Murasugi-Przytycki upper bound for alternating links; the paper confirms it for several classes but does not establish it in general.