Quantum advantage threshold conjecture for grouped MAJORITY

Let $n$ players be divided into two groups of sizes $n-k$ and $k$, with $k$ fixed. Let $V_Q^*$ and $V_C^*$ denote the optimal quantum and classical values, respectively, for the $MAJORITY$ game. Quantum advantage threshold conjecture. There is a threshold $n_k$ such that quantum advantage disappears for

$$n \geq n_k.$$

The paper gives exact values of $n_k$ for several even values of $k$ and discusses asymptotic convergence for odd $k$, but does not provide a closed-form general solution.