Multiplicative functions which are additive on triangular numbers

Fix $k \ge 3$. If a multiplicative function $f$ satisfies $$f(x_1+x_2+\dots+x_k) = f(x_1) + f(x_2) + \dots + f(x_k)$$ 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 $k=2$.