$k$-additive uniqueness of the set of squares for multiplicative functions

P. V. Chung showed that there are many multiplicative functions $f$ which satisfy $f(m^2+n^2)=f(m^2)+f(n^2)$ for all positive integers $m$ and $n$. In this article, we show that if more than $2$ squares in the additive condition are involved, then such $f$ is uniquely determined. That is, if a multiplicative function $f$ satisfies $$f(a_1^2+a_2^2+\cdots +a_k^2)=f(a_1^2)+f(a_2^2)+\cdots +f(a_k^2)$$ for arbitrary positive integers $a_i$, then $f$ is the identity function. In this sense, we call the set of all posotive squares a \emph{$k$-additive uniqueness set} for multiplicative functions.