Liu–You–Tang–Liu conjecture on the exponential reduced Sombor index

Let $G=(V(G),E(G))$ be a graph, let $d(v)$ denote the degree of $v$, and let $T$ be a chemical tree, meaning that $d_T(v)\leq 4$ for every vertex $v$. Write ${\cal CT}_n$ for the set of chemical trees of order $n$, and define the exponential reduced Sombor index by

$$e^{SO_{red}}(G)=\sum_{uv\in E(G)}e^{\sqrt{(d(u)-1)^2+(d(v)-1)^2}}.$$

Liu–You–Tang–Liu's conjecture. For every $T\in {\cal CT}_n$ with $n\geq 5$,

$$e^{SO_{red}}(T)\leq \left\{\begin{array}{ll} \frac{2}{3}(n+1)e^3+\frac{1}{3}(n-5)e^{3\sqrt{2}}, & n\equiv 2 (\!\!\!\!\mod 3);\\ \frac{1}{3}(2n+1)e^3+\frac{1}{3}(n-13)e^{3\sqrt{2}}+3 e^{\sqrt{13}}, & n\equiv 1 (\!\!\!\!\mod 3);\\ \frac{1}{3}ne^3+\frac{1}{3}(n-9)e^{3\sqrt{2}}+2 e^{\sqrt{10}}, & n\equiv 0 (\!\!\!\!\mod 3). \end{array}\right.$$

The conjecture concerns the maximum exponential reduced Sombor index among chemical trees of a fixed order. The paper's abstract says that its proposed bound is negative, while the supplied status evidence distinguishes it from another conjecture that was solved and says this one remains open; the exact relationship between that evidence and the negative result should be checked.