One of my crazy ideas after getting up in the morning :-)
Consider the SPG 1.e4 f6 2.Qh5+. At least it's guaranteed that
this position occurs for the last time in this game.
Still, a LPG (longer proof game :-) 1.Sc3 Sc6 2.Sb1 Sb8 3.e4 f6 4.Qh5+
is possible, as the knight moves are reversible.
So, an UPG (unique proof game) is only possible where there are
no reversible moves throughout the game. E.g. assume the knights are moas...
still not enough, wQ and bK can lose tempi.
Offhand example where an UPG works: Haan. So assume the position is
after 1.Nf3, hole g1, then indeed this position can occur *only*
in the game 1.Nf3.