Sun–Tauraso q-congruences for central q-binomial coefficients

Let $p$ be a prime, $a\geq1$, let $\Phi_{p^a}(q)$ be the $p^a$th cyclotomic polynomial, and let $\left(\frac{p^a}{3}\right)$ denote the Legendre symbol. Sun–Tauraso q-conjecture. One has

$$\sum_{k=0}^{p^a-1}q^k{2k\brack k}_q\equiv\left(\frac{p^a}{3}\right)q^{\left\lfloor\frac{p^a}{2}-\left(\frac{p^a}{3}\right)\frac{p^a}{6}\right\rfloor+\left\lfloor\frac{p^a}{3}\right\rfloor p^a}\pmod{\Phi_{p^a}^2(q)},$$

and, if $p\neq3$,

$$\sum_{k=0}^{p^a-1}q^{2k+1}{2k\brack k}_q\equiv\left(\frac{p^a}{3}\right)q^{\left(\left\lfloor\frac{p^a+1}{3}\right\rfloor+\left(\frac{p^a}{3}\right)\right)p^a}\pmod{\Phi_{p^a}^2(q)}.$$

These are proposed q-versions of the cited Sun–Tauraso congruence.