Fix an integer $d \geq 2$ and for every real number $x$ let $M_d(x)$ be number of $d \times d$ matrices $(a_{ij})$ satisfying: every $a_{ij}$ is a positive integer, the product of every row does not exceed $x$, and the product of every column does not exceed $x$.

I'm looking for a good upper bound for $M_d(x)$ as $x \to +\infty$.

If we forget about the condition on the columns, since it is well known that the number of $d$-tuples $(b_1, \dots, b_d)$ of positive integers satisfying $b_1 \cdots b_d \leq x$ is $\ll_d x (\log x)^{d - 1}$ (a generalization of Dirichlet divisor problem), we get the upper bound $$M_d(x) \ll_d x^d (\log x)^{d(d-1)}.$$

In the special case $d = 2$, we have that $a_{12}, a_{21} \leq \min(x / a_{11}, x / a_{22})$ and consequently $$M_2(x) \leq \sum_{a_{11}, a_{22} \leq x} \min\left(\frac{x}{a_{11}}, \frac{x}{a_{22}}\right)^2 \ll \sum_{a_{11} \leq a_{22} \leq x} \left(\frac{x}{a_{22}}\right)^2 \ll x^2 \log x ,$$ which is a better upper bound than the general one given in the above paragraph. However, I have no idea of how to generalize this trick to $d \geq 3$ (if possible).

Has this problem been studied before? Do you have any idea/suggestion about it?