Multiplicative and Jordan multiplicative maps on structural matrix algebras

Let $M_n$ denote the algebra of $n\times n$ complex matrices and let $\mathcal{A}\subseteq M_n$ be an arbitrary structural matrix algebra, i.e. a subalgebra of $M_n$ that contains all diagonal matrices. We consider injective maps $\phi :\mathcal{A}\to M_n$ that satisfy the condition $$\phi (X\bullet Y)=\phi (X)\bullet \phi (Y),\text{for all }X,Y\in \mathcal{A},$$ where $\bullet$ is either the standard matrix multiplication $(X,Y)\mapsto XY$, the Jordan product $(X,Y)\mapsto XY+YX$, or the normalized Jordan product $(X,Y)\mapsto \frac{1}{2}(XY+YX)$. We show that all such maps $\phi$ are automatically additive if and only if $\mathcal{A}$ does not contain a central rank-one idempotent. Moreover, in this case, we fully characterize the form of these maps.