A sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues

We prove a sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues of bounded Lipschitz domains in $\mathbb R^N$. More precisely, if $N\ge2$ and $\Omega\subset\mathbb R^N$ is a bounded Lipschitz domain, then $$
\prod_{j=1}^N \sigma_j(\Omega)\le \frac{\omega_N}{|\Omega|}, $$ where $0=\sigma_0(\Omega)<\sigma_1(\Omega)\le\sigma_2(\Omega)\le\cdots$ are the Steklov eigenvalues of $\Omega$, and $\omega_N$ denotes the volume of the unit ball in $\mathbb R^N$. This extends the convex-domain theorem of Henrot, Philippin, and Safoui to arbitrary bounded Lipschitz domains, and in particular settles the remaining higher-dimensional case of a problem posed by Henrot.