The conjectured asymptotic value of the minimum degree threshold $d_n$

Let $d_n$ denote the parameter defined in the paper’s minimum-degree theorem for friends-and-strangers graphs. Minimum-degree threshold conjecture. We have

$$d_n=\dfrac{3}{5}n+O(1).$$

This conjecture predicts the asymptotic value of the minimum-degree threshold in the general setting and is presented as a desired improvement of the paper’s estimates; its resolution is not supplied in the source.