Spectral extension property for unitizations of semisimple Banach algebras

Let $(A,\|\cdot\|)$ be a non-unital, semisimple Banach algebra, and let $A_e$ denote its unitization. Say that $A$ has the spectral extension property (SEP) when every norm on $A$ with the relevant spectral-extension property also has it after passing to the unitization. Spectral extension conjecture. If $A$ has SEP, then $A_e$ has SEP for semisimple Banach algebras.

This conjecture concerns whether SEP is preserved by unitization in the semisimple case. The paper's main theorem gives equivalent conditions characterizing this preservation, but the supplied text does not establish the conjecture independently; its resolution should therefore be checked against the theorem and its hypotheses.