P2006T Flight Instructor reported the engine would not start back up after shutting down the engine for flight training. After running through the checklist multiple times and consulting the Assistant Chief and Maintenance; the pilot performed an air turnback.

Narrative

During a multi-engine training flight in the vicinity of ZZZ; the student pilot and I; the CFI; intentionally shut down the right engine to perform single-engine operations. Everything was performed according to the manufacturer's procedure and we were maneuvering around the area; teaching the student how an aircraft performs on one engine while conducting the lesson as scheduled. After maneuvering I instructed the student to restart the engine. At this point; the student followed the checklist thoroughly; without missing any item - carburetor heat as required; electrical fuel pump on; checking the fuel quantity and fuel selector; field off; ignition switches on; throttle lever to idle position; propeller lever full forward; and starting the engine. The engine did not start as it should. At this point I got the flight controls and ran through the checklist multiple times; read the Airplane Flight Manual to see if there was another source of information to try and restart the engine; turned the emergency battery on; and after this didn't work I decided to contact the flight school's Assistant Chief. With him we went through what could have been the probable causes; discussed new courses of action - checking the circuit breakers; increasing the airspeed; verifying every switch was in the correct position - and decided to contact Maintenance. After speaking with Maintenance I made the decision as the pilot in command to [request priority handling] and come back to ZZZ for a single-engine landing with the appropriate procedures for the maneuver. Upon contacting ZZZ Tower I immediately [requested priority handling]; squawked; and answered ATC regarding the number of souls on board; 2 souls on board; and fuel on board - 2 hours of fuel. We got cleared to land on Runway XXR and I performed a single-engine landing as described by the procedure. After landing; runway was vacated and the airplane stopped; I shut it down and evacuated the airplane safely while waiting for ground assistance.