A fifth-power q-congruence for the convolution in Theorem 4

Let $n$ be a positive odd integer, and let $c_q(k)$ be the sequence defined in Theorem 4. Let $[n]$ denote the $q$-integer and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomial. The conjectural generalization.

$$\sum_{k=0}^{n-1}\sum_{j=0}^{k}c_q(j)c_q(k-j) \equiv q^{1-n}[n]^2+\frac{(n^2-1)(1-q)^2}{12}q[n]^4 \pmod{[n]^2\Phi_n(q)^3}.$$

The modulus contains the fifth power of a cyclotomic polynomial after accounting for the $q$-integer factor, and the source describes this type of congruence as difficult and unsolved.