Unemployment questionnaires

So my partner was fired from his job, and as a result is in the process of filing for unemployment while he continues to look for other jobs. He continues to claim weekly (New York), and this past week he received a questionnaire in the mail regarding him quitting a past employer. This was a part time job he was working while also doing his full time. He decided to quit this job because it took time out of full-time job duties, and he felt secure at his full-time job. 8 months later, the full time job fires him.

I know that he has to tell the truth about this, however, can he be denied unemployment benefits because of this? Nowhere in the questionnaire does it ask if he would take the job back, just if he did quit and had he secured employment elsewhere.