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

For a fixed integer $k \ge 3$, we study the multiplicative functions $f\colon\mathbb{N}\to\mathbb{C}$ satisfying $$f\Bigl(\sum_{i=1}^{2k} x_i^2\Bigr) = \sum_{j=1}^{k} f\bigl(x_{2j-1}^2 + x_{2j}^2\bigr)$$ 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 $f(2)\neq 0$ 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 $n > 2k + 21$.