I was hoping someone can help me work out potential for collider bias in the following scenario:

Exposure (E) potentially causes one of the model covariates (A) used to adjust the exposureoutcome relationship. (I recognize this is already problematic in the sense that the covariate may be on the causal pathway to the outcome, but here I want to focus for potential for collider bias).

Other variables (B) are associated with the covariate in question, perhaps causally to some extent. They are also associated with the outcome (D).These variables are also included as covariates in the model.
If covariates B are not completely included in the model, I think we have potential for collider bias. Am I correct?
If the covariates B are completely included in the model, assuming perfect adjustment, I think this theoretically removes collider bias. Is this correct?
Thanks very much