A matching-indexed doubling counterexample

Let $H(n)$ count labeled real Hadamard matrices of order $n$. An injective construction gives $H(2m)\ge(2m-1)!!\,H(m)^2$: given a perfect matching of the $2m$ row positions and Hadamard matrices $A,B$ of order $m$, the two rows of the $i$th pair are $(A_{i,\cdot},B_{i,\cdot})$ and $(A_{i,\cdot},-B_{i,\cdot})$, and the matching, $A$ and $B$ are recovered from the output, so no assumption about automorphism groups or equivalence classes enters. Iteration yields $H(2^k)\ge 2^{2^k(k-1)(k-2)/8}$ for $k\ge2$. Consequently no absolute constant $C$ satisfies $H(n)\le2^{Cn\log_2 n}$ at every order divisible by four. This refutes the counting conjecture attributed to Ferber, Jain and Zhao; it does not address existence at every admissible order.