The non-isolated order-six Hadamard matrix conjecture

Let $H_1$ be any complex Hadamard matrix of order $6$, not equivalent to the isolated matrix $S_6$. Let $g_1$ be the function defined by the source's equation, and let $\rho$ be any permutation of the vector $(1,1,1,-1,-1,-1)$. The non-isolated order-six Hadamard matrix conjecture. Then

$$g_1(\rho)=0.$$

This conjecture was proposed as a possible step toward proving that a complete system of mutually unbiased Hadamard matrices does not exist in dimension $6$. The source does not establish the assertion; its status is therefore open.