Recall that the multipartition function $p_k(n)$ counts the number of $k$-tuples of partitions $\lambda^1,\ldots,\lambda^k$ of numbers $a_1,\ldots,a_k$ with $a_1+\cdots+a_k=n$. It has a generating function: $$\sum_np_k(n)x^n=\prod_{n\geq 1}(1-x^n)^{-k}$$ Asymptotics of $p_k(n)$ are known for $k$ fixed and $n\to\infty$, see Theorem 4 in Murty.

Is there a known asymptotic formula for $p_k(n)$ which is uniform in both variables as $k,n\to\infty$?

If this is not known, I would also be interested in partial results (upper/lower bounds).