The correct answer is A. is.
The sentence “However honest he is, I do not trust him” is a statement of fact. The speaker is not expressing uncertainty about the man’s honesty, but rather their lack of trust in him. The use of the present tense verb “is” indicates that the speaker believes that the man is honest at the time of speaking.
The other options are all modal verbs, which express possibility or necessity. The verb “may” expresses possibility, the verb “might” expresses possibility or uncertainty, and the verb “could” expresses possibility or ability. In this context, none of these modal verbs would make sense. The speaker is not expressing uncertainty about the man’s honesty, nor are they saying that he is capable of being honest. They are simply stating that they do not trust him.
Therefore, the correct answer is A. is.