# Is every quasi-nilpotent element in a C$^*$-algebra a norm-limit of nilpotent elements?

Let $$A$$ be a C$$^*$$-algebra. I have seen theorems either stating or implying that if $$A$$ is the algebra of bounded linear operators on a separable Hilbert space (Herrero et al.), or the Calkin algebra, or an AF-C$$^*$$-algebra (Skoufranis), then every quasi-nilpotent element in $$A$$ is a norm-limit of nilpotent elements. Is the general case known: is every quasi-nilpotent element in a C$$^*$$-algebra a norm-limit of nilpotent elements?