The sup-norm problem for automorphic cusp forms of $\mathrm{P}\mathrm{G}\mathrm{L}(n,\mathbb{Z}[i])$

Let $\phi$ be an $L^2$-normalized Hecke–Maaß cusp form for ${\mathrm{P}\mathrm{G}\mathrm{L}}_n(\mathbb{Z}[i])$ on the locally symmetric space $X:={\mathrm{P}\mathrm{G}\mathrm{L}}_n(\mathbb{Z}[i])\mathrm{\backslash }{\mathrm{P}\mathrm{G}\mathrm{L}}_n(\mathbb{C})\mathrm{/}{\mathrm{P}\mathrm{U}}_n$. If $\mathrm{\Omega }$ is a compact subset of $X$, then we prove the bound $\mathrm{\parallel }\phi {\mathrm{\mid }}_{\mathrm{\Omega }}{\mathrm{\parallel }}_{\mathrm{\infty }}{\ll }_{\mathrm{\Omega }}{\lambda }_{\phi }^{n(n-1)\mathrm{/}4-\delta }$ for some $\delta >0$ depending only on $n$, where ${\lambda }_{\phi }$ is the Laplace eigenvalue of $\phi$.