Exact count conjecture for time-invariant Nash equilibria

Let $Q:(0,\infty)\to\mathbb{N}$ denote the maximum cardinality of a set of mutually shift-inequivalent time-invariant Nash equilibria for the game $\operatorname{Standard}(x)$, and let $\lambda$ be the endpoint parameter for the admissible Mina-margin interval. Exact-count conjecture.

$$Q(x)=2\quad\text{when }x\in(\lambda,\lambda^{-1}),\qquad Q(x)=1\quad\text{when }x\in\{\lambda,\lambda^{-1}\}.$$

This refines the preceding existence result by asserting uniqueness up to shift in the interior and at both endpoints; the supplied text gives no resolution.