Conjectured constrained moment-matching lower bound in the interpolation regime

Let $p$ be the norm parameter, let $L\geq 1$ be the number of matched moments, and let $M_p(\epsilon,L)$ be the constrained moment-matching quantity. For $\epsilon$ satisfying

$$0<\epsilon\lesssim \frac{1}{L},$$

Constrained moment-matching lower-bound conjecture.

$$M_p^{1/p}(\epsilon,L)\gtrsim \begin{cases} \dfrac{\epsilon^{p/2}}{L^{1-p/2}} & \text{if } 1\leq p<2,\\ \dfrac{\epsilon^{2k/p}}{L^{1-2k/p}} & \text{if } 2k<p<2(k+1)\text{ with }k\in\mathbb{Z}_+. \end{cases}$$

Such a bound would yield lower bounds on the critical separation for non-smooth $\ell_p$-norm testing in the interpolation regime. The conjecture is presented as the missing constrained moment-matching lower bound and is not established in the supplied text.