Chebyshev formulae for Kadar–Yu Gram determinant polynomials

Let $l$ and $l'$ be integer parameters, and let $P_n^{(\lambda)}(\alpha)$ and $C^{(\lambda)}(\alpha)$ be the polynomials appearing in the factorization of the Kadar–Yu Gram determinants. The displayed formulae define these factors for the partitions $(l'+2)$ and $(1^{l+2})$, together with their associated Chebyshev expansions.

Chebyshev formula conjecture. Formulae for $C^{(l'+2)}(\alpha)$, $C^{(1^{l+2})}(\alpha)$, $P^{(l'+2)}_{l'+4+k}(\alpha)$, and $P^{(1^{l+2})}_{l+4+k}(\alpha)$ hold for all $l\geq -1$ and $l'\geq 0$.

These identities extend computations verified only over finite ranges of the parameters and would provide uniform formulae for the corresponding Gram determinants.