A strengthened convolution q-congruence for Theorem 8

Let $n$ be a positive odd integer, and let $c_q(k)$ be the sequence defined in Theorem 8 by

$$c_q(k)=[4k+1]\frac{(q;q^2)_k^6}{(q^2;q^2)_k^6}q^k.$$

Let $[n]$ denote the $q$-integer and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomial. The strengthened convolution q-congruence. Modulo $[n]^2\Phi_n(q)^3$,

$$\sum_{k=0}^{n-1}\sum_{j=0}^{k}c_q(j)c_q(k-j) \equiv q^{1-n}\left([n]^2+\frac{(n^2-1)(1-q)^2}{12}[n]^4\right) \left(\sum_{k=0}^{(n-1)/2}\frac{(q;q^2)_k^4}{(q^2;q^2)_k^4}q^{2k}\right)^2.$$

This is stated as a stronger version of Theorem 8 and is left as an apparent conjecture.