Conjecture on the maximal number of terms in the averaged Baker–Campbell–Hausdorff expansion

Let $Q_{L,d}$ denote the maximal possible number of terms in the relevant expansion at tensor level $L$ for dimension $d$, and let $R_L$ be the tensor-product polynomial in the averaged Baker–Campbell–Hausdorff expansion defined by

$$R_L(\cdot)=\left(\sum\limits_{\substack{k=1,\dots,L\\ k\text{ odd}}}\mathsf{aBCH}(\mathbf{b},\mathbf{c}^{(i)})\right)_L.$$

Term-count conjecture. For $d\geq L$, the maximal possible number of terms satisfies $Q_{L,d}=Q_{L,L}$ and coincides with the number of terms obtained by expanding $R_L$ in the averaged Baker–Campbell–Hausdorff formula.

The conjecture concerns the combinatorial complexity of tensor expansions arising from the barycenter construction in free nilpotent Lie groups. The examples given for $L=3$ and $L=4$, and the supplementary computation for $L=5$, provide initial term counts but do not establish the general equality.