The universal tunnel-number bound conjecture for Dehn surgeries

Let $K'$ be a knot in $S^3$, let $M=K'(p/q)$ be a non-trivial Dehn surgery on $K'$, and let $g$ be the Heegaard genus of $M$. Let $t(K')$ denote the tunnel number of $K'$.

Universal tunnel-number bound conjecture. There is a function

$$w_{TN}:\mathbb{N}\to\mathbb{N}$$

such that

$$t(K')\leq w_{TN}(g).$$

The preceding corollary proves such a linear bound under hyperbolicity, non-integrality, and a boundary-slope exclusion. The conjecture asks for a universal bound without those additional restrictions.