Unlabelled inversion diameter conjecture for complete graphs

Let $K_n$ be the complete graph on $n$ vertices, and let $\mathcal{U}(K_n)$ be its unlabelled inversion graph, obtained by identifying vertices of the inversion graph corresponding to isomorphic orientations. Unlabelled inversion diameter conjecture.

$$\operatorname{diam}(\mathcal{U}(K_n))\leqslant n-\lceil\log(n+1)\rceil.$$

The bound is motivated by the known upper bound for the eccentricity of the transitive tournament in the unlabelled inversion graph. The source gives asymptotic lower and upper bounds but does not establish this conjectured inequality.