Optimality conjecture for knots obtained by braid substitution

Let $L$ be a link with a free decomposing surface $S$, and replace the trivial braids in $N(S)\cap L$ by an arbitrary collection of braids. The resulting link is obtained by braid substitution; in the constructions of Propositions 4.5 and 4.6, denote the resulting knots by $K$. Optimality conjecture. There exist knots $K$ obtained from the links of Propositions 4.5 and 4.6 by braid substitutions which are optimal. The statement concerns whether braid substitutions can produce knots realizing the relevant tunnel-number bounds optimally; the supplied text gives no resolution, so its status remains open.