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Let $A$ be a unital Banach algebra. It is well-known that, under the hypothesis that all closed left ideals in $A$ are algebraically finitely-generated as left ideals, necessarily $A$ is a finite dimensional algebra. We investigate whether the same conclusion holds under the weaker assumption that all maximal left ideals of the unital Banach algebra $A$ are algebraically finitely-generated. We prove that this is true whenever $A$ is a commutative unital Banach algebra and whenever $A$ is a unital $C^*$-algebra. We then try to prove the conjecture for every Banach algebra of the form $B(E)$; this is the Banach algebra of all bounded linear operators on a Banach space $E$. We succeed for many, but not all, Banach spaces $E$. We have to look at some recently developed "exotic" Banach spaces $E$.