One property of a planar curve whose convex hull covers a given convex figure

In this note, we prove the following conjecture by A. Akopyan and V. Vysotsky: If the convex hull of a planar curve $\gamma$ covers a planar convex figure $K$, then $\operatorname{length}(\gamma) \geq \operatorname{per} (K) - \operatorname{diam} (K)$. In addition, all cases of equality in this inequality are studied.