Bounds for the binomial Fisher limiting prior

Determine the limsup and liminf of the equilibrium prior P^*_N(x_A,x_B) for the binomial Fisher games {N,x_A,x_B} as N \to \infty with 0 < x_A < x_B < 1, and prove that P^*_N(x_A,x_B) does not converge but has finite upper and lower bounds given by explicit closed-form expressions.

Background

In the binomial Fisher limit {N,x_A,x_B}, the sampling distributions are binomial and the symmetric equilibrium yields a prior P^*_N(x_A,x_B) that appears not to converge as N grows. Numerical evidence and heuristic derivations indicate oscillatory behavior with apparent envelope bounds.

The conjecture asserts the existence of finite limiting superior and inferior values of P^*_N(x_A,x_B) and proposes explicit formulas for these bounds (referenced in the text), but a rigorous proof establishing both the non-convergence and the precise bounding values remains open.