Multiplicative functions additive on partitions of $2k$ nonzero squares

For a fixed integer $k\ge 3$, we study the multiplicative functions $f\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{C}$ satisfying $$f\left(\sum _{i=1}^{2k}{x}_i^2\right)=\sum _{j=1}^{k}f(x_{2j-1}^2+x_{2j}^2)$$ for all positive integers $x_1,\dots ,x_{2k}$. This extends a theorem of Park on sums of two nonzero squares, which established the $k=2$ case. For $k=3$ and $k=4$, we prove that every such $f$ with $f(2)\mathrm{\ne }0$ is the identity function on $\mathbb{N}$. For $k\ge 5$, we show that such a function $f$ must be either the identity function on $\mathbb{N}$, or $f(n)=0$ for all $n>2k+21$.