Cabling conjecture for the primitive bridge spectrum of pretzel knots

Let $K_j=K_j(p_1,p_2,\ldots,p_j)$ be a pretzel knot with stair-step primitive bridge spectrum. Let $T_{m,n}$ be an $(m,n)$-torus knot, and set

$$p=\sum_{i=1}^j p_i.$$

If $K={\mathsf {cable}}(T_{m,n},K_j)$ is a $T_{m,n}$ cable of $K_j$, then the primitive bridge-spectrum conjecture.

$${\mathbf {\hat b}}(K)=(mj,m(j-1),\ldots,3m,2m,\min\{m,\lvert mp-n\rvert\},0).$$

The conjecture formalizes the observed degeneration in the last nonzero bridge number under cabling. The authors indicate that methods may work for $n=3$, but that cases with $n\geq 4$ face difficulties from closed incompressible surfaces in Montesinos-knot exteriors.