Prime-order divisor conjecture for automorphisms of Kimura Hadamard matrices

Let $H$ be a Kimura Hadamard matrix (KHM) of order $4(2k)+4$, and let $(R,S)\in\operatorname{Aut}(H)$ be an automorphism of prime order $p$. Prime-order divisor conjecture. Then

$$p=2\quad\text{or}\quad p\mid k.$$

The conjecture is suggested by all examples presented in the paper. Under the preceding block-monomial automorphism conjecture, the authors explain that primes $p>3$ dividing an automorphism-group order satisfy $p\leqslant k$, and reduce the action to the individual blocks, but they do not establish the asserted divisibility in general.