Block-monomial automorphism conjecture for Kimura Hadamard matrices

Let $H$ be a Kimura Hadamard matrix (KHM) of order $8k+4$. An automorphism is a pair $(R,S)$ acting on $H$ by the relation $RH=HS$. The known automorphisms have block-diagonal form

$$R=\operatorname{diag}(R_1,R_2),\qquad S=\operatorname{diag}(S_1,S_2),$$

where $R_1,S_1\in\operatorname{Mon}_4(\{\pm1\})$ and $R_2,S_2$ are $8k\times 8k$ block-monomial matrices with blocks of size $2k\times 2k$, each itself block-monomial with sub-blocks of size $k\times k$. Block-monomial automorphism conjecture. Every automorphism $(R,S)$ of a KHM has this block-monomial form. The conjecture is motivated by the uniform structure of all automorphisms found in the known constructions and examples; the paper does not report a resolution.