Morimoto's conjecture on Heegaard genus degeneration of connected sums

A knot exterior $X$ is the complement of an open tubular neighborhood of a knot, and $g(X)$ denotes its Heegaard genus. An exterior admits a primitive meridian if it has a minimal-genus Heegaard surface separating it into a compression body and a handlebody, with a compressing disk in the handlebody and a vertical annulus in the compression body whose boundary data intersect once as specified in the definition. For knots $K_1$ and $K_2$, let $X_1$ and $X_2$ be their exteriors, and let $X$ be the exterior of their connected sum $K_1\#K_2$. Morimoto's conjecture. If $K_1$ and $K_2$ are knots in $S^3$, then

$$g(X) < g(X_1) + g(X_2)$$

if and only if $X_1$ or $X_2$ admits a primitive meridian. The conjecture characterizes precisely when Heegaard genus is strictly subadditive under connected sum for knots in $S^3$; the preceding result establishes that a primitive meridian is sufficient, while the claimed necessity is the unresolved part for arbitrary knots in $S^3$.