Doubling conjecture for Sylow-2 sandpile groups of consecutive hypercubes

For $n\geq 1$, let $Q_m$ denote the $m$-dimensional hypercube and let $\operatorname{Syl}_2(K(Q_m))$ denote the Sylow-$2$ component of its sandpile group. Doubling conjecture.

$$\operatorname{Syl}_2(K(Q_{2^n}))\cong \operatorname{Syl}_2(K(Q_{2^n-1}))^2\times \mathbb{Z}/(2^{2^n+n-1}\mathbb{Z}).$$

Equivalently, apart from the displayed top cyclic factor, the cyclic-factor multiplicities for $Q_{2^n}$ are obtained by doubling those for $Q_{2^n-1}$. The statement is presented as a remaining conjecture; its reference to the top factor relies on results discussed earlier in the paper.