Conjectural braid-index formulas for exceptional Type 3 pretzel links

Let $\mathcal{L}$ be a Type 3 pretzel link, and let $\textbf{b}(\mathcal{L})$ denote its braid index. For positive parameters, consider

$$\mathcal{L}\in P_3(\mu_1;-(\mu_1+1)\vert 2\alpha_1,\ldots,2\alpha_{\kappa^+};0)$$

with $\mu_1>1$ and $1<\min\{\alpha_j\}\le \mu_1$, or

$$\mathcal{L}\in P_3(1;-3\vert 2\alpha_1,\ldots,2\alpha_{\kappa^+-2},4,2;0)$$

with $\alpha_j\ge 2$ for $1\le j\le \kappa^+-2$. For negative parameters, consider

$$\mathcal{L}\in P_3(\nu_1+1;-\nu_1\vert 0;-2\beta_1,\ldots,-2\beta_{\kappa^-})$$

with $\nu_1>1$ and $1<\min\{\beta_i\}\le \nu_1$, or

$$\mathcal{L}\in P_3(3;-1\vert 0;-2\beta_1,\ldots,-2\beta_{\kappa^--2},-4,-2;0)$$

with $\beta_i\ge 2$ for $1\le i\le \kappa^--2$.

Conjectural Type 3 braid-index formulas. In the first positive case, $\textbf{b}(\mathcal{L})=2+\sum\alpha_j$; in the second positive case, $\textbf{b}(\mathcal{L})=1+\sum\alpha_j$. In the first negative case, $\textbf{b}(\mathcal{L})=2+\sum\beta_i$; and in the second negative case, $\textbf{b}(\mathcal{L})=1+\sum\beta_i$.

These formulas are proposed as the concluding conjecture of the authors' study of Type 3 pretzel links, based on computational results and the proof of one related case. The stated braid indices are not established in the source, so the conjecture remains open.