Randić-index inequality for triangle-free graphs

Let $G$ be a triangle-free graph, let $R(G)$ be its Randić index, and let $s^+(G)$ be the sum of the squares of the positive adjacency eigenvalues. Triangle-free Randić conjecture.

$$R(G)\ge\sqrt{s^+},$$

with equality if and only if $G$ is a complete bipartite graph. The paper presents this as a conjecture following the connected-graph Randić bounds and supplies no proof of the triangle-free assertion.