Multiplicative functions commutable with sums of squares

Let $k$ be an integer greater than or equal $4$. We show that if a multiplicative function $f$ satisfies $f(x_1^2 + x_2^2 + \dots + x_k^2) = f(x_1)^2 + f(x_2)^2 + \dots + f(x_k)^2$ for all positive integers $x_i$'s, then $f$ is the identity function.

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