XQAOA level-1 limiting performance conjecture for the Binary Paint Shop Problem

Consider Binary Paint Shop Problem instances encoded as Ising models, and let $\Delta_C$ be the number of paint swaps. Let XQAOA$_1$ denote the level-1 XQAOA ansatz, with its parameters optimised by gradient-based methods. XQAOA level-1 performance conjecture. The limiting expected paint-swap ratio is approximately $0.357$:

$$\lim_{n\rightarrow\infty}\mathbb{E}\left[\frac{\Delta_C}{n}\right]\approx 0.357.$$

This conjecture is motivated by numerical experiments showing that the XQAOA$_1$ distributions remain stationary across the tested problem sizes, but no asymptotic proof is given.