On the equation of the $\rho$-orthogonal additivity

We solve a conditional functional equation of the form $$x{\perp }^{\rho }y\Rightarrow f(x+y)=f(x)+f(y),$$ where $f$ is a mapping from a real normed linear space $(X,\Vert ·\Vert )$ with $\text{dim}X\ge 2$ into an abelian group $(G,+)$ and ${\perp }^{\rho }$ is a given orthogonality relation associated to the norm.