Expected number of hyperplane cuts to partition a $d$-dimensional hypercube of side length $n$ into pieces of unit volume

Question

I came up with the following question and am not certain of its solution. Has this problem or problems like it be tackled before?

Let $H_d$ be the $d$-dimensional hypercube $[0,n]^d \subset \mathbb{R}^d$. Define a cut to be a hyperplane which intersects $H_d$. In expectation, how many uniformly sampled cuts are needed so they partition $H_d$ into pieces that all have unit volume or less?

I speculate the answer is $O(n^d \log(n))$.

The one dimensional version of the question is posted separately here, where the answer is $\Theta(n\log(n))$