This question is in some sense dual to this question.
Let $A$ be a unital $C^*$-algebra. It is known that if $u$ and $v$ are unitaries in $A$ with
$$\|u-v\|<2,$$
then $u$ and $v$ are homotopic through a path of unitaries.
Question: Does a similar statement hold for invertible elements? More precisely, if $a$ and $b$ are invertible in $A$, does there exist $\delta>0$ such that $a$ and $b$ are homotopic through invertible elements whenever $$\|a-b\|<\delta?$$ (In particular, what if $b$ is the identity?)