Multivariate analogue of Carlitz's identity for $\sum_{m\ge 0}\prod_i [m+1]_{q_i}^{n_i}T^m$

I'm interested in a generalization of the Carlitz's classical identity

$\sum_{m\ge 0}[m+1]_q^n T^m = \frac{\displaystyle\sum_{\sigma\in S_n} q^{\operatorname{maj}(\sigma)}T^{\operatorname{des}(\sigma)}} {\displaystyle\prod_{j=0}^n(1-q^jT)}.$

I am interested in a multivariate version of the left-hand side. Let $ n_1,\ldots,n_r\ge 0$, and consider $ F_{\mathbf n}(\mathbf q,T) = \sum_{m\ge0} \prod_{i=1}^r [m+1]_{q_i}^{\,n_i}T^m.$

A straightforward expansion shows that $F_{\mathbf n}$ is rational, with denominator dividing

$\prod_{k_1=0}^{n_1}\cdots\prod_{k_r=0}^{n_r} \left(1-q_1^{k_1}\cdots q_r^{k_r}T\right).$

My question is whether there is a known combinatorial analogue of Carlitz's formula for this series.

More precisely:

• Is there a natural class of permutations, multiset permutations, colored permutations, or partitions for which the numerator of $F_{\mathbf n}(\mathbf q,T)$ is a generating polynomial?
• Is there a multivariate statistic $\operatorname{maj}_1,\ldots,\operatorname{maj}_r,\operatorname{des}$ such that one obtains an identity of the form $F_{\mathbf n}(\mathbf q,T)=\frac{\displaystyle\sum_{\pi}q_1^{\operatorname{maj}_1(\pi)}\cdots q_r^{\operatorname{maj}_r(\pi)}T^{\operatorname{des}(\pi)}}{ \displaystyle \prod_{k_1=0}^{n_1}\cdots\prod_{k_r=0}^{n_r} (1-q_1^{k_1}\cdots q_r^{k_r}T) },$ possibly after cancellations or with a slightly different natural denominator?
• More generally, is this series already covered by a known multivariate Carlitz identity?

This is clearly the Hadamard product of the ($r$) series $\sum_{m\ge 0}[m+1]_{q_i}^{n_i}T^m$, but the poor compatibility between the Hadamard product and the ordinary Cauchy product seems to make a direct derivation quite difficult.

I would also be interested in references where this type of multivariate generating function is studied.

Thanks a lot!