A q-supercongruence from Rahman's summation formula

Let $n equiv 1\pmod{4}$ be a positive integer. Define the $q$-integer $[m]=(1-q^m)/(1-q)$, the $q$-shifted factorial $(a;q)_k=\prod_{j=0}^{k-1}(1-aq^j)$, and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomial. The q-supercongruence conjecture.

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

and

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

These strengthen the preceding q-congruences and are motivated by Rahman's summation formula; their validity remains open in the source.