On the degeneration of tunnel numbers under a connected sum

We show that, for any integer n ≥ 3, there is a prime knot k such that (1) k is not meridionally primitive, and (2) for every m-bridge knot kʹ with m ≤ n, the tunnel numbers satisfy t(k#kʹ) ≤ t(k). This gives counterexamples to a conjecture of Morimoto and Moriah on tunnel numbers under a connected sum and meridionally primitive knots.