The correct answer is D. No improvement.
The sentence “She would not go to his house unless I go with her” is grammatically correct and conveys the intended meaning. The use of the past tense “would go” is correct because it is used to express a hypothetical situation. The use of the modal verb “would” indicates that the action is not certain to happen. The use of the negative “not” indicates that the speaker is sure that the woman will not go to the man’s house unless the speaker goes with her. The use of the prepositional phrase “unless I go with her” indicates that the woman will only go to the man’s house if the speaker is with her.
The other options are incorrect because they do not convey the intended meaning. Option A, “would go,” is incorrect because it does not express the speaker’s certainty that the woman will not go to the man’s house unless the speaker goes with her. Option B, “went,” is incorrect because it is in the past tense, which is not appropriate for a hypothetical situation. Option C, “shall go,” is incorrect because it is in the future tense, which is not appropriate for a situation that is not certain to happen.