Multiplicative functions which are additive on triangular numbers

Fix $k\ge 3$. If a multiplicative function $f$ satisfies $$f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k)$$ for arbitrary positive triangular numbers $x_1,x_2,\dots ,x_k$, then $f$ is the identity function. This extends Chung and Phong’s work for $k=2$.