Daniel–Jolliffe conjecture on the 2-elementary divisors of the hypercube adjacency matrix

Let $Q_n$ be the $n$-cube graph, let $A$ be its adjacency matrix, and suppose that $n$ is even. A nonzero integer has exact $2$-power divisor $2^r$ if it is divisible by $2^r$ but not by $2^{r+1}$. The multiplicity of $2^i$ as a $2$-elementary divisor of $A$ is the number of eigenvalues of $A$ whose exact $2$-power divisor is $2^{i+1}$.

Daniel–Jolliffe conjecture. For every $i$, the multiplicity of $2^i$ as a $2$-elementary divisor of $A$ equals the number of eigenvalues of $A$ with exact $2$-power divisor $2^{i+1}$.

The eigenvalues of $A$ are $n-2\ell$ with multiplicity $\binom{n}{\ell}$ for $0\leq\ell\leq n$. The conjecture concerns the unresolved 2-primary part of the Smith normal form when $n$ is even; the corresponding elementary divisors for odd primes were determined, while the cited source did not compute the 2-elementary divisors.