Suppose a student currently doing Ph.D in an institute wants to leave the institute and apply abroad for Ph.D. Should he/she mention the name of the institute he/she wants to leave? If the universities call the current PI, they will never get a positive feedback since the current institute will never want to lose their student. But the student wants to mention that he/she has work experience. What is to be done in this situation? Can the US universities reject the application based on the current PI's feedback?