C172 Check Pilot reported conducting a private pilot checkride and getting in close proximity to another aircraft that was on an IFR flight plan.

Narrative

I was conducting a private pilot checkride. The applicant was wearing foggles and I was giving him radar vectors when I spotted the other aircraft at our altitude; 5000 ft.; a few miles away. We were heading west. The other aircraft was moving south. We were converging and so I asked the applicant to give me a constant airspeed descent. I observed that the other airplane began a climb and so I asked the applicant to level off at 4800 ft. I began giving position reports. The other airplane did not make any radio calls. My plan was to stay higher; 4800 - 5000 ft.; because eventually that applicant would be doing a simulated emergency descent. I noticed the other airplane begin a turn towards the west and so I asked my applicant to turn to a heading of 090 to avoid a conflict. What I didn't realize was that the other aircraft was initiating a 360 turn to the right and so even though we were heading east the other aircraft circled around and ended up over the top and a little in front of us.I asked the applicant to maintain his altitude while the other aircraft passed in front and above us. I observed that we were around 400 ft. below. Once the conflict was over the other aircraft came on the frequency and asked us what were were doing. I explained we were doing a checkride but I did not want to engage the other pilot over the frequency. He claimed we were as close as 300 ft. which I disagreed with. As soon as we landed I looked up the tail number and discovered that they were probably on an IFR flight plan. In retrospect; once I saw the aircraft after it circled and was in front of us I should have taken the controls and initiated an aggressive descent. However; at the time; I didn't perceive the aircraft to be a threat. Because my applicant was in the middle of a test I wanted to keep the test as calm as possible and so I allowed the aircraft to pass in front of us while I kept a watchful eye on it.