Morimoto's conjecture on tunnel number and primitive meridians

Let $K$ and $K'$ be knots, let $K\#K'$ denote their connected sum, and let $t(K)$ denote the tunnel number. A knot admits a primitive meridian when its exterior has a minimal genus Heegaard splitting containing a vertical essential annulus and a meridian disk whose boundaries intersect once. Morimoto's conjecture. If

$$t(K \# K') \leq t(K)+t(K'),$$

then $K$ or $K'$ admits a primitive meridian.

This is the converse of the known implication that a primitive meridian gives the displayed tunnel-number inequality. The supplied source does not itself state a resolution for this formulation, so its database status remains open.