A q-analogue of the Van Hamme–Sun supercongruence equivalence

Let $n\equiv 1\pmod{4}$ be a positive integer. Define the $q$-integer $[m]=(1-q^m)/(1-q)$, the $q$-shifted factorial $(x;q)_k=(1-x)(1-xq)\cdots(1-xq^{k-1})$, and let $\Phi_n(q)$ denote the $n$-th cyclotomic polynomial. q-analogue conjecture.

$$\sum_{k=0}^{n-1}(-1)^k[6k+1]\frac{(q;q^2)_k^3}{(q^4;q^4)_k^3}q^{3k^2} \equiv \sum_{k=0}^{n-1}(-1)^k[8k+1]\frac{(q;q^4)_k^3}{(q^4;q^4)_k^3}q^{2k^2+k} \pmod{[n]\Phi_n(q)^3}.$$

This is a proposed $q$-analogue of an equivalence between two supercongruences for primes congruent to $1$ modulo $4$; its validity for all positive integers $n\equiv1\pmod4$ remains open.