Large-parameter $c$-stability for a family of 2-bridge knots

Let

$$r_m=[\underbrace{2,2,\dots,2}_{m-1},2k,\underbrace{-2,-2,\dots,-2}_{m}].$$

Large-parameter stability conjecture. There exists a positive integer $N_m$ such that: if $m$ is even, then $K(r_m)$ is $c$-stable for $k\geq N_m$; and if $m$ is odd, then $K(r_m)$ is $c$-stable for $k\leq-N_m$. The claim predicts eventual unit-circle location of the Alexander-polynomial zeros in this explicit family.