Test worth the effort? Start the engine on vegatable oil. Get it warmed up and shut down. Restarts easily the viscosity of the hot vegatable oil remains higher than hot diesel fuel. Many ex WVO burners had this problem when put back on diesel fuel in the day.Cooling down the injection pump with water is not recommended.

Also although I do not think it is your case. Excessive engine blowby will shut down the injection pump. At hot idle engine should not launch a loose oil filler cap.

Could also be an obstructed crankase breathing system. You could also be building vacuum in the fuel tank with a blocked tank vent hose. Leave the fuel cap off to test for this. It is probably the injection pump but you want to make sure.

Poster already mentioned about the bad ignition switch valve. Most are long shots but takes no time to clear them.

At least you know what you are doing.In the day young fellows would burn WVO and when the problems started would ask for help. The problem was most of them at best had only changed a flat tire. That was about the total extent of their mechanical experience on cars.