A q-congruence for the central trinomial coefficient

Let $p$ be a prime with $p\geq 5$. Define the $q$-analogue of the trinomial coefficient by

$$\left({n\choose j}\right)_q=T_1(n,j,q)=\sum_{k=0}^{n}q^{k(k+j)}{n\choose k}_q{n-k\choose k+j}_q,$$

where $[n]_q=(1-q^n)/(1-q)$ and $\lfloor x\rfloor$ denotes the integral part of the real number $x$. The q-congruence conjecture. One has

$$\left(2p\choose p\right)_q\equiv\left(2\left\lfloor\frac{p+3}{6}\right\rfloor+p\right)(q^p-1)+2\pmod{[p]_q^2}.$$

This is proposed as a $q$-analogue of the preceding trinomial congruence, motivated by numerical calculation; its validity is left as a problem for further research.