Lipschitz-free spaces over products of sequences

We answer positively a question of Aliaga and show that for any nonconstant real polynomial $p$, the Lipschitz-free space over $\{(p(n), p(m)):n, m\in \mathbb{N}\}$ is isomorphic to $\mathcal{F}(\mathbb{Z}^2)$. We in fact show more generally that if $d\in \mathbb{N}$, $q\in \mathbb{Z}_{\geq 0}$, and $((a_n^{(i)})_{n=1}^\infty)_{i=1}^d$, $((b_m^{(j)})_{m=1}^\infty)_{j=1}^q$ are sequences with $0<a_1^{(i)}<a_2^{(i)}<\cdots$, $0<b_1^{(j)}<b_2^{(j)}<\cdots$, $a_n^{(i)}\to \infty$ as $n\to \infty$, $\frac{a_{n+1}^{(i)}}{a_n^{(i)}}\to 1$ as $n\to \infty$ and $\underset{m\to \infty}{\liminf}{\frac{b_{m+1}^{(j)}}{b_m^{(j)}}}>1$ for all $i, j$, then the Lipschitz-free space over the product of these $d+q$ sequences is isomorphic to $\mathcal{F}(\mathbb{Z}^d)$.