The strengthening of Lemma 3

Let $n$ and $s$ be the parameters occurring in Lemma 3, and let $\Phi_{n^s}(q)$ be the corresponding cyclotomic polynomial. The strengthening conjecture. The q-congruence in Lemma 3 holds modulo

$$\Phi_{n^s}(q)^2.$$

The source states that this would strengthen Lemma 3; the lemma's formula is not included in the supplied context.