Semicontinuity conjecture for Nash equilibrium types in generic two-action games

Let $({\mathcal A},G,V)$ be a generic two-action game, with $m$ players, Nash equilibrium set ${\mathcal N}$, and $C_l$ denoting the set of points of $C_l$-type. For each $d\in\{0,1,\ldots,m\}$, one has

$$\sum_{l=0}^d |{\mathcal N}\cap C_l|\leq !m+\sum_{l=1}^d \begin{pmatrix}m\\l\end{pmatrix}\cdot 2^{l-1}\cdot !(m-l).$$

Semicontinuity conjecture. The displayed inequality holds for every generic two-action game and every $d\in\{0,1,\ldots,m\}$.

This strengthens the bounds known for product two-action games and for generic two-action games sufficiently close to product games. It is motivated by the observation that, under larger deformations, the $C_l$-type of a Nash equilibrium may change; the conjecture asserts that the cumulative numbers of equilibria of each type nevertheless satisfy the product-game bounds.