Conference-construction maximal-parameter conjecture for complex Hadamard matrices

Let $D$ be an arbitrary dephased, symmetric complex Hadamard matrix arising from the conference matrix construction, of order $N$. Apply Construction by repeatedly selecting suitable pairs of rows and corresponding columns and introducing one free parameter at each step; the construction can introduce at most $\frac{N}{2}-1$ parameters. Conference-construction maximal-parameter conjecture. Construction leads to Hadamard matrices after each step, and, for $N\geq 14$, the maximum number $\frac{N}{2}-1$ of parameters can be introduced. The conjecture concerns the sufficiency of the two suitability conditions and the attainment of the maximal parameter count in this construction; the source reports examples for $D_{10}$ and $D_{14}$ but no general proof.