Conjecture on non-stabilized boundary stabilizations for the T(7,17,2,10m−4) family

Let $K_m=T(7,17,2,10m-4)$, and let $(V,W)$ be the unique minimal genus-two Heegaard splitting of its exterior $E(K_m)=S^3-N(K_m)$. Let $(V',W')$ be the boundary stabilization of $(V,W)$, which is a genus-three Heegaard splitting. The non-stabilized boundary-stabilization conjecture. The splitting $(V',W')$ is non-stabilized. The construction gives weakly reducible genus-three splittings, and the non-$5cmu$-primitive properties established for this family rule out the obvious stabilization mechanism; whether these splittings are non-stabilized is left as a conjecture.