GA pilot reported fuel would not transfer from the auxiliary tank to the right main tank which caused an inflight shutdown.

Narrative

I conducted a thorough pre-flight to include visibly checking fuel levels and gauges. I calculated fuel endurance based on the available fuel and the scheduled flight for the day. I started up Aircraft X at XA:27 at ZZZ. I flew the airplane to ZZZ1 where we were picking and dropping the banners. That leg took 0.5 hours. I started up at XB:43 at ZZZ1 in order to pick the banner. I picked the banner at XB:53. I then navigated over the freeway westbound en route to an event taking place. The airplane has 40 gallons in the main wing fuel tanks. There is an auxiliary fuel tank behind the pilot seat that pumps fuel into the right main tank in order to allow the fuel to reach the motor. The aux fuel pump is activated by holding a switch in the up position until a green indicator light illuminates on the panel. The company procedure is to activate the aux pump when the right main fuel gauge indicates that half its capacity has been used up. I turned on the fuel pump in accordance with this procedure. I visually verified that the indicator light was illuminated. I continued to monitor the indicator light and the right main fuel gauge. The aux tank also has a fuel quantity gauge. I monitored that as well. As fuel is pumped from the aux tank to the right main tank; that aux gauge should show a decrease; and the right main fuel gauge should hold steady at half full until the aux tank is emptied and the fuel level is burned below half in the right main tank. As I monitored the fuel system over time; I began to realize that the aux tank quantity indication; the right main tank quantity indication; and the fuel totalizer gauge were indicating a potential issue. The aux tank was still indicating full when it had enough time to be reduced from full and the right main was not holding half full (it was prematurely below half). I said something out of concern to my coworker and senior pilot over the air-to-air radio frequency. The senior pilot; responded that the maintenance had been done to include inspecting the aux fuel system. He told me I should have no reason to be concerned. The right main continued to decrease; and the aux tank quantity maintained the full indication. Approximately 20 minutes later; I radioed again that the situation seemed to be deteriorating. This time; the senior pilot's tone was more annoyed and he stated that I had a similar concern last year; but they had already addressed it so I didn't need to keep bringing it up. He said that maybe the fuel gauge was malfunctioning; or maybe the angle of attack was causing a faulty indication for the right main fuel gauge or the aux tank gauge; or both. I allowed this external pressure to affect my initial decision after the first call on the radio to not divert sooner. After this second radio call; I chose to start heading back early to ZZZ1 where we were conducting banner operations. I initiated a climb while trying to manage the fuel burn as efficiently as possible while monitoring the engine and oil temp gauges. The engine failure occurred during this climb out heading northeast bound en route to ZZZ1. The airplane was at approximately 1;300 feet indicated. There was a field behind me that I had identified in case of an issue. I was just north of that field when the engine quit. I did not want to jettison the banner over the residential neighborhood; so I immediately turned 180 degrees back to the field and dropped the banner. The winds were strong from the north according to a recent weather broadcast at nearby ZZZ; so I knew I needed to continue the turn to land north in the field. I took a photo of the tach at XE:28 when I shutdown right after touchdown in the field.