Morimoto's tunnel-number conjecture for connected sums of knots

Let $K_1,K_2\subset S^3$ be knots, let $K_1\#K_2$ denote their connected sum, and let $t(K)$ be the tunnel number of a knot $K$. For a knot $K$, write $E(K)$ for its exterior. A primitive meridian in a Heegaard splitting $(V_1,V_2)$ of $E(K_1)$ is a meridian curve $\mu\subset\partial E(K_1)$ isotopic to a curve $\mu^*$ on the Heegaard surface $\partial V_2$ such that $\mu^*$ intersects an essential disk $D\subset V_2$ in a single point.

Morimoto's conjecture. The knots $K_1\subset S^3$, $K_2\subset S^3$, and $K_1\#K_2$ satisfy

$$t(K_1\#K_2)\leq t(K_1)+t(K_2)$$

if and only if either $E(K_1)$ or $E(K_2)$, say $E(K_1)$, has a minimal-genus Heegaard splitting $(V_1,V_2)$ with $\partial E(K_1)\subset V_1$ and a primitive meridian.

The conjecture gives a criterion for when the usual upper bound for the tunnel number of a connected sum can be improved by one. The source notes that it was known for the special class of smallish knots, namely knots that do not contain essential meridional surfaces; its general status is not specified here.