Kogiso–Miyamoto–Ren–Wakui–Yanagawa arithmetic conjecture for prime denominators

Let $p$ be an odd prime integer, and let $a,b$ be positive integers coprime to $p$. The associated $q$-deformed rational-number polynomials are denoted by $\mathcal{S}_{a/p}(q)$ and $\mathcal{S}_{b/p}(q)$. Arithmetic conjecture. One has

$$\mathcal{S}_{\frac{a}{p}}(q)=\mathcal{S}_{\frac{b}{p}}(q)$$

if and only if $ab\equiv -1 \pmod{p}$ or $a\equiv b \pmod{p}$. This conjecture concerns when two polynomials associated with fractions having the same prime denominator coincide; the source presents it as the main motivation of the paper, based on examples and prior master’s theses.