Morimoto's conjecture on Heegaard genus under connected sum

Let $K_1$ and $K_2$ be knots in the 3-sphere $S^3$. Write $K_1\#K_2$ for their connected sum, let $t(K)$ denote the tunnel number of a knot, let $E(K)$ denote its exterior, and let $g(E(K))$ denote the Heegaard genus of its exterior. A knot $K$ in a closed orientable manifold admits a $(g,n)$ position if it lies with $n$ arcs simultaneously parallel into the boundary of each handlebody of some genus-$g$ Heegaard splitting.

Morimoto's conjecture. The inequality

$$g(E(K_1\#K_2))<g(E(K_1))+g(E(K_2))$$

holds if and only if, for $i=1$ or $i=2$, $K_i$ admits a $(t(K_i),1)$ position.

Morimoto proved the converse implication when $K_1$ and $K_2$ are m-small knots. The paper exhibits counterexamples in general, so the conjecture is refuted; the later prime-knot restriction is a distinct refinement.