Nonzero equivalence conjecture for negative scoring positions

Let $\mathcal{PT}_{-x}$ be the set of scoring-play game positions in which every node has value $x>0$ and the initial score is $0$. Write $G\cong 0$ for equivalence to the zero game and $G=0$ for equality with the zero game. Nonzero equivalence conjecture. For all $G\in\mathcal{PT}_{-x}$, if $G\not\cong 0$ then $G\neq 0$. This follows a discussion of whether scoring games can behave like normal-play games, but the source provides no argument resolving the stated implication.