Let $A$ be a unital $C^*$-algebra and $I$ a proper ideal of $A$.
Let $h$ be an idempotent in $A/I$, and let $\tilde{h}$ be a lift of $h$ to $A$. Then a direct calculation shows that $$e^{2\pi i\tilde{h}}\in I^+,$$ where we consider the unitization $I^+$ as a subalgebra of $A$ in the natural way. This can be verified using the series for the exponential function.
In the case that $\tilde{h}$ is itself an idempotent in $A$, we would have $$e^{2\pi i\tilde{h}}=1_{I^+}$$ on the dot.
Question: Is there a constant $\delta>0$ (independent of $A$ and $I$) such that if the lift $\tilde{h}$ satisfies $\|\tilde{h}^2-\tilde{h}\|<\delta$, we have $$e^{2\pi i\tilde{h}}\sim_h 1_{I^+},$$ where $\sim_h$ means homotopic through a path of invertibles in $I^+$?
Remark: I guess it suffices to show the existence of a $\delta$ such that starting from any "$\delta$-close idempotent" lift, one can produce an idempotent lift.