Moment lower-bound conjecture for weighted sums of exponential random variables

Let $q>0$ be real, let $a_1,\dots,a_n$ be non-negative real numbers satisfying

$$\sum_{i=1}^n a_i^2=1,$$

and let $X_1,\dots,X_n$ be independent identically distributed standard exponential random variables. Define $\rho(j,q)$ as the moment value associated with the configuration having $j$ nonzero equal weights, as used in the source. Moment lower-bound conjecture.

$$\mathbb{E}\big(a_1X_1+\cdots+a_nX_n\big)^q\geqslant\min\\{\rho(1,q),\dots,\rho(n,q)\\}.$$

The conjecture extends the behavior observed in the preceding analysis from the displayed parameter choices to every real $q>0$. The supplied statement does not define $\rho(j,q)$ explicitly or report a resolution.