Fiendish knot conjecture on tunnel-number additivity

Let $K_1,K_2\subset S^3$, write $K=K_1\#K_2$, and let $E(K_i)$ denote the exterior of $K_i$. Let $t(K_i)$ denote the tunnel number, and call a Heegaard splitting minimal genus if its Heegaard surface has the smallest possible genus. A minimal-genus splitting has a primitive meridian when it admits the corresponding primitive-meridian configuration. Fiendish knot conjecture.

$$t(K)=t(K_1)+t(K_2)+1$$

if and only if both $E(K_1)$ and $E(K_2)$ do not have minimal-genus Heegaard splittings with primitive meridians. This conjecture characterizes the stated one-unit failure of tunnel-number additivity in terms of the absence of primitive meridians; the source attributes it to the case of fiendish knots and leaves it unresolved.