A prime-power $q$-binomial congruence

Let $p$ be a prime, let $n$ be a power of $p$, and define the $q$-integer by

$$[n]_q=\frac{1-q^n}{1-q}.$$

For Gaussian binomial coefficients, write ${a\brack b}_q$. Prime-power $q$-binomial congruence. Then

$$\sum_{k=0}^{n-1}q^{(n-k)^2}{n+k\brack k}_q^2{n-1\brack k}_q^2\equiv q^{(n-1)^2}[n]_q\pmod{[p]_{q^{n/p}}^2}.$$

This is a $q$-analogue of congruences for binomial sums at prime-power parameters; the excerpt does not state whether it has been proved or remains open.