Approximate Nash equilibrium from a mean-field optimal-control minimizer

Let $(m,m\mathbf v)$ be a minimizer of the dual optimal-control problem, and let $\eta$ be the induced probability measure on continuous trajectories $\Gamma=C([0,T];\mathbb T^N)$, with $e_t\#\eta=m(t)$. Let $\bar\beta:\Omega\times\mathbb T^N\to\Gamma$ be a measurable map representing the disintegrated trajectory measures, so that

$$\int_\Gamma G(\gamma)\,d\eta(\gamma)=\int_{\Omega\times\mathbb T^N}G(\bar\beta(\omega,x))\,d\mathbb P(\omega)m_0(dx).$$

The induced strategy $\bar\beta$ is an open-loop strategy, meaning that it depends only on the random parameter $\omega$ and the initial position $x_0$. Approximate Nash-equilibrium conjecture. Under certain assumptions on $k$, for every $\epsilon>0$ and all sufficiently large $n$, the strategy $\bar\beta$ satisfies

$${\bf J}_i^n(\beta^i,(\bar\beta)_{j\neq i})\geq {\bf J}_i^n(\bar\beta,(\bar\beta)_{j\neq i})-\epsilon$$

for every strategy $\beta^i$ of player $i$ and every $i=1,\ldots,n$. This asserts that the mean-field optimal-control solution provides an approximate Nash equilibrium for the associated finite-player differential game; the precise assumptions on $k$ and the proof of the approximation are not supplied in the stated conjecture.