I don't remember seeing "whomever" before, so I hope you don't mind me trying to understand where it is used.
I see that it is used for the object of a verb, and whoever for the subject of a verb, but I am a bit confused here, because here it is the object of "downvote", but the subject of "doubts", so I am confused.
In the sentence "I will look at whomever Bob greets.", I think I understand why it is "whomever", and in the sentence "Whoever enters the room next, they will encounter a surprise.", I think I understand why it is "whoever" and not "whomever", but in the case of your sentence, I don't know what the rule is.
Could you (or anyone else) please explain it to me?
It's an interesting corner case. I believe it's "I will look at whomever Bob greets." (as you note) and "I will look at whoever greets Bob." The object of the preposition is the phrase "whoever greets Bob," so subject-to-a-verb wins.
According to [0], it seems that "whoever" would be correct here.
> Rule 1. The presence of whoever or whomever indicates a dependent clause. Use whoever or whomever to agree with the verb in that dependent clause, regardless of the rest of the sentence.
> Examples:
> Give it to whoever/whomever asks for it first.
> He asks for it first. Therefore, whoever is correct.
