# Lifting quasi-nilpotent elements in C$^*$-algebras

Let $$A$$ be a C$$^*$$-algebra with closed two-sided ideal $$I$$. Set $$B=A/I$$ and let $$\pi:A\to B$$ be the quotient map. Suppose that $$b\in B$$ is quasi-nilpotent. Does there exist quasi-nilpotent $$a\in A$$ such that $$\pi(a)=b$$?

I am rather certain this is unknown. The issue is that procedures we know to lift the relation $$a^{n+1}=0$$ do not mesh well with the $$a^n=0$$ case. This means writing the given quasinilpotent as a limit of nilpotents does not seem to help.