Suppose we want to estimate \begin{equation} y = \alpha_0 + x_1 \beta + c'\gamma + \varepsilon \end{equation} where $\beta$ is the variable of interest and $c$ is a vector of controls. We suspect $E[x\varepsilon]\neq0$ but we have an instrument $z$ such that $E[z\varepsilon]=0.$ My question is - as long as IV is valid - should we care that there could be an endogenous control $c_1$ such that $E[c_1\varepsilon]\neq0$? Why or why not?
My hunch is that it doesn't because as long as IV is valid (it satisfies inclusion-exclusion principle) the correlation between controls and unobservables should not lead to the inconsistency of the estimate of $\beta$.