A strengthened parametric q-congruence for Theorem 3

Let $n$ be a positive odd integer, and let $a$ be an indeterminate. Write $[n]$ for the $q$-integer and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomial. Let the congruences referenced as

$$and$$

be the congruences stated in Theorem 3 and its parametric form. The strengthened q-congruence. The congruence

should hold modulo $[n]^2(1-aq^n)(a-q^n)$; in particular,

should hold modulo $[n]^2\Phi_n(q)^2$. This is presented as a numerical strengthening of Theorem 3, with no proof supplied.