On multiplicative functions which are additive on positive cubes

Let $k\ge 3$. If a multiplicative function $f$ satisfies $$f(a_1^3+a_2^3+\cdots +a_k^3)=f(a_1^3)+f(a_2^3)+\cdots +f(a_k^3)$$ for all $a_1,a_2,\dots ,a_k\in \mathbb{N}$, then $f$ is the identity function. The set of positive cubes is said to be a $k$-additive uniqueness set for multiplicative functions. But, the condition for $k=2$ can be satisfied by infinitely many multiplicative functions. Besides, if $k\ge 3$ and a multiplicative function $g$ satisfies $$g(a_1^3+a_2^3+\cdots +a_k^3)=g(a_1{)}^3+g(a_2{)}^3+\cdots +g(a_k{)}^3$$ for all $a_1,a_2,\dots ,a_k\in \mathbb{N}$, then $g$ is the identity function. However, when $k=2$, there exist three different types of multiplicative functions.