The Spectral extension property in the unitization of Banach Algebras

Let $A$ be a non-unital Banach algebra and let $A_e=A\oplus \mathbb{C}1$ be the unitization of $A$. It is true that if $A_e$ has the spectral extension property (SEP), then $A$ has the same. Does the converse hold? In this paper, we give some necessary as well as some equivalent conditions.