The symmetry-symbol conjecture for normalized non-exceptional PPSs

Let $v$ be odd. A normalized propus parameter set (PPS) is a parameter set $(v;k_1,k_2,k_3,k_4;\lambda)$ for which the associated PDF has $X_2=X_3$; it is non-exceptional when the four block sizes $k_i$ are not all equal. A PDF has symmetry symbols $(s{**})$ or $({**}s)$ when, respectively, its first or fourth block is symmetric and the other indicated blocks have arbitrary symmetry symbols.

Symmetry-symbol conjecture. For each normalized and non-exceptional PPS $(v;k_1,k_2,k_3,k_4;\lambda)$ there exist PDFs with symmetry symbols $(s{**})$ and $({**}s)$.

The conjecture is implicit in earlier work and had been verified there for odd $v\leq 53$; the source considers it in the context of constructing symmetric Hadamard matrices and lists it for the subsequent range of odd parameters.