There have been a couple of questions on Ramanujan's $\tau$ function.

Lehmer's conjecture for Ramanujan's tau function

The Vanishing of Ramanujan's Function tau(n)

A $q$-analogue is given by $$ \tau_q(n) = \sum_{|\lambda|=n}\prod_{(i,j)} \frac{[5-h(i,j)][5+h(i,j)]}{[h(i,j)]^2} $$

Here $[k]=\frac{q^k-q^{-k}}{q-q^{-1}}$. The sum is over all partitions of $n$. The product is over all boxes in the diagram of $\lambda$ and $h(i,j)$ is the hook length of the box $(i,j)$.

This is a $q$-analogue because (despite appearances) it is a Laurent polynomial in $q$ and substituting $q=1$ gives $\tau(n)$.

It would be ridiculous to ask if this is $q$-holonomic.

Is this $q$-analogue known? and does it have any significance in number theory?