Weak reducibility conjecture for additive tunnel numbers

Let $K_1$ and $K_2$ be knots in $S^3$, and write $K=K_1\#K_2$. Let $E(K)$ denote the exterior of $K$, and let $t(K)$ denote its tunnel number. A Heegaard splitting is weakly reducible if it admits disjoint compressing disks on opposite sides, and a splitting has minimal genus when its Heegaard surface has the smallest possible genus. Weak reducibility conjecture. If

$$t(K_1\#K_2)=t(K_1)+t(K_2),$$

then $E(K)$ has a minimal-genus Heegaard splitting that is weakly reducible. This proposes weak reducibility as a structural consequence of additive tunnel number, extending the examples and results discussed in the source; the conjecture is presented as open.