Banach algebras satisfying $pq=qp=q \Rightarrow \|q\|\leq\|p\|$ for all idempotents $p$ and $q$

Question

Let $A$ be a Banach algebra with the property $\big(q=pq=qp \Rightarrow \|q\|\leq \|p\|\big)$ whenever $p,q\in A$ are idempotents.

Is there a term coined to the algebras with this property in the literature?

For an example, $\ell^2$ with pointwise addition and multiplication has this property, whereas its unitization $\ell^2\oplus\mathbb{C}$ does not.