Unimodality conjecture for coefficients of constructed polynomial Pythagorean triples

A polynomial solution $(\mathcal{A},\mathcal{B},\mathcal{C})$ is constructed for each standard Pythagorean triple, where $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are polynomials in $q$. A sequence of real numbers is unimodal if it increases, not necessarily strictly, to a maximum and then decreases monotonically. Unimodality conjecture. The sequences of coefficients of the polynomials $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ constructed in this article are unimodal. The authors cannot guarantee this additional property; unimodality is generally difficult to prove. The analogous property for $q$-deformed rational numbers was conjectured previously and proved in the cited work.