The stronger edge bound conjecture for stress-independent graphs

Let $G=(V,E)$ be a $d$-stress-independent graph, where $d$-stress independence is the stress condition defined in the paper. Assume that $G$ has sufficiently many vertices.

Stronger edge bound conjecture. The number of edges satisfies

$$|E|\leq (d+1)|V|-(d+1)^2.$$

The theorem immediately preceding this conjecture gives the weaker bound $(d+1)|V|-\binom{d+2}{2}$ for suitable subgraphs. The conjecture proposes a stronger asymptotic upper bound, but does not specify the threshold for “sufficiently many” vertices.