Murasugi sum conjecture for the braid index of special alternating links

Let $L$) be the planar Murasugi sum of special alternating links $L_1,\ldots,L_n$. Murasugi sum conjecture. Then

$$b(L)-1=\sum_{i=1}^n\bigl(b(L_i)-1\bigr).$$

This conjecture was proposed in connection with the analogous proved equality for the algebraic crossing number. It would imply the stated classification of alternating $\mathcal{F}$-links when combined with the results preceding it.