Hriňáková–Knor–Škrekovski strong conjecture on variable Wiener and Szeged indices

For a connected graph $G$, let

$$W^{\alpha}(G):=\sum_{u,v\in V(G)}d(u,v)^{\alpha},\qquad \operatorname{Sz}^{\alpha}(G):=\sum_{uv\in E(G)}\bigl(n_u(v)n_v(u)\bigr)^{\alpha},$$

where $d(u,v)$ is the distance between vertices $u$ and $v$, and $n_u(v)$ and $n_v(u)$ count vertices closer to $u$ and $v$, respectively, among the endpoints of the edge $uv$. Hriňáková–Knor–Škrekovski strong conjecture. For every non-complete graph $G$ there is a constant $\alpha_G\in(0,1)$ such that

$$\operatorname{Sz}^{\alpha}(G)>W^{\alpha}(G)\quad\text{if }\alpha>\alpha_G;$$ $$\operatorname{Sz}^{\alpha}(G)=W^{\alpha}(G)\quad\text{if }\alpha=\alpha_G;$$ $$\operatorname{Sz}^{\alpha}(G)<W^{\alpha}(G)\quad\text{if }\alpha<\alpha_G.$$

This conjecture asserts a single critical exponent governing the transition between the variable Wiener and variable Szeged indices for every non-complete connected graph.