Conjecture on the odd moments of singular series differences

Let $k\geqslant 1$. For each finite set $\mathcal D$ of distinct positive integers, let $\mathfrak S_0(\mathcal D)$ denote the modified singular series, and define

$$R_k(h):=\sum_{\substack{d_1,\ldots,d_k\\1\leqslant d_j\leqslant h\d_j\ \operatorname{distinct}}}\mathfrak S_0(\{d_1,\ldots,d_k\}).$$

Let $\mu_{2k}=(4k)!/(2^{2k}(2k)!)$, and set

$$r_{2k+1}:=\tfrac32(2k+1)k\mu_{2k}.$$

Odd-moment conjecture. As $h$ tends to infinity,

$$R_{2k+1}(h)\sim (-1)^{k-1}r_{2k+1}h^k(\log h)^{k+1}.$$

This refines the conjectural order of magnitude for odd moments of the modified singular series. The case $k=1$ is established by the theorem in the paper, while the asserted asymptotic for all $k\geqslant 1$ remains conjectural.