That sounds highly possible they way the detent system works. Although were talking two different shifting forks timed by the drum. (this means if the detent spring was broken and the drum rolled the first gear fork would be moving away from first as the second gear fork move in towards second gear. same but reversed when down shifting) So if first and second to be engaged at once it could also be your dealer finds a bent fork if he splits it. First thing I would do is scope the detent spring and see if it is still together and mounted. this can be done before you even pull the radiator. now looks the snap ring on fifth so the gear floats could put you in a position where what ever gear your in the transmission will probably lock up or sound like a bucket of gravel seems more likely. Since fifth would be floating on the shaft.
Still I would stick a scope up the water pump hole in the clutch cover first. This way I could decide which way I am going to have to go with the repair. scope shot of good detent spring below thru the water pump hole cheap Harbor Freight Scope.
what year R3T do you have?