Rainbow connecting $2$-colorings of super-Dirac graphs

Let $G$ be a graph with minimum degree $δ(G)\ge|V(G)|/2$. Can we color the edges of $G$ with red and blue so that every pair of non-adjacent vertices is connected by a path consisting of exactly one red edge and one blue edge? We provide an affirmative answer to this question for a class of graphs that are ``close'' to a complete balanced bipartite graph or the disjoint union of two cliques of the same order. Surprisingly, our methods extend to a much broader class of graphs with minimum degree slightly above $|V(G)|/2$. Furthermore, we answer an asymptotic version of this question in full, proving that every graph $G$ satisfying $δ(G)\ge(|V(G)|-1)/2$ has a $2$-edge-coloring such that almost all pairs of vertices are connected by a rainbow path. In addition, we propose a number of related open problems.