Edge bound conjecture for minimally ${\cal R}_d$-connected graphs

Let $G=(V,E)$ be a minimally ${\cal R}_d$-connected graph, meaning that $G-e$ is not ${\cal R}_d$-connected for every edge $e\in E(G)$. The graph $K_{d+2}$ denotes the complete graph on $d+2$ vertices.

Edge-bound conjecture. One has

$$|E|\leq (d+1)|V|-\binom{d+2}{2},$$

with equality if and only if $G=K_{d+2}$.

The cases $d=1,2$ were settled in the cited work of Jear, while the general-dimensional assertion is the subject of the conjecture.