Conjecture on reversible Hadamard difference sets in $C_{2^n}\times C_{2^n}$

Let $G_n=C_{2^n}\times C_{2^n}$. A reversible Hadamard difference set is a reversible difference set with Hadamard parameters. The results concern the total number of such difference sets of types A and B.

Reversible Hadamard difference set enumeration conjecture. The total number of reversible Hadamard difference sets in $G_n$ of type A is

$$(2^{2^n-2})(2^n+1),$$

and the total number of those of type B is

$$(2^{2^n-2})(2^n-1).$$

Hence the total number of reversible Hadamard difference sets in $G_n$ is

$$2^{2^n+n-1}.$$

This conjecture was prompted by a question of James A. Davis and John B. Polhill about the total number of reversible Hadamard difference sets. The paper reports complete enumeration of regular nontrivial canonical partial difference sets of even size for $n\leq 9$, but the computational results for $n=10,11$ are likely incomplete; the stated formulas are suggested by those enumeration results.