Hölder control of diameter by swept-area distance

Let $K$ be a knot type, let $Y_{\Lambda}(K)$ be the representatives at ropelength level $\Lambda$, and let $d_{\Lambda}^{K}$ be the swept-area pseudometric. For representatives $\gamma_0,\gamma_1$ in the same admissible component, write $\operatorname{diam}(\gamma)$ for the Euclidean diameter of $\gamma$.

Diameter variation conjecture. For each ropelength level $\Lambda$, there should exist constants $C(\Lambda)>0$ and $\alpha>0$ such that

$$|\operatorname{diam}(\gamma_1)-\operatorname{diam}(\gamma_0)| \le C(\Lambda)\\,d_{\Lambda}^{K}(\gamma_0,\gamma_1)^\alpha.$$

The exponent $\alpha=1/2$ is a natural first candidate. A proof or counterexample would clarify whether diameter and compression radius can provide robust lower bounds for swept-area distance.