The conjecture on boundary stabilization of non-primitive knots

Let $K\subset S^3$ be a knot that is not $\gamma$--primitive, and let a minimal genus Heegaard splitting of its exterior $E(K)$ be given. A boundary stabilization is the genus-increasing amalgamation described in the source. Boundary-stabilization conjecture. A boundary stabilization of the minimal genus Heegaard splitting of $E(K)$ is non-stabilized. The source explains that boundary stabilization is stabilized when the knot is $\gamma$--primitive, while the asserted non-stabilization in the non-primitive case remains unresolved.