Linear lower bound for compressing-disc boundary lengths in handlebodies

Let $H^i_0$ and $H^i_1$ be the handlebodies referred to in the source's preceding remark, and let $g$ denote the relevant genus. Linear boundary-length conjecture. There exists $C>0$ such that, after possibly switching $H^i_0$ and $H^i_1$, every compressing disc $D$ for $H^i_0$ satisfies

$$\operatorname{length}(\partial D)>Cg.$$

This is proposed as an approach to the question of geometric control of handlebodies; the supplied text gives no evidence of a proof or disproof.