Are the dates correct?
It seems you joined the new company before being relieved from the old company.
So from 05-Jun-17 to 13-Jun-17 you were an employee of both companies. (Salaries from both employers)
Joining a new company without being relieved from old company is a conflict of interest. The above determines dual employment from your side.
New Company joining date should be always after the last company relieving date.
So the concern of the background agency is genuine.
14th August 2017 From India, Mumbai