Spectrum preserving linear mappings for scattered Jordan-Banach algebras

Given two semisimple complex Jordan-Banach algebras with identity $A$ and $B$, we say that $T$ is a spectrum preserving linear mapping from $A$ to $B$ if $T$ is surjective and we have $\operatorname{Sp}(Tx)=\operatorname{Sp}(x)$, for all $x\in A$. We prove that if $B$ is a scattered Jordan-Banach algebra, then $T$ is a Jordan isomorphism.