The strong chromatic index $χ'_s(G)$ is the smallest number of colours needed to colour the edges of a graph $G$ so that any two edges at distance at most $2$ receive different colours. Using the \emph{local flag algebra} framework introduced in a companion paper, we prove $χ'_s(G) \leq 1.73\,Δ(G)^2$ for every graph $G$ of maximum degree $Δ(G)$, $χ'_s(G) \leq 1.6255\,Δ(G)^2$ for every bipartite $G$, and $χ'_s(G) \leq 1.6633\,Δ_A(G)\,Δ_B(G)$ for every bipartite $G$ of side maximum degrees $Δ_A(G), Δ_B(G)$ with rational $Δ_B(G)/Δ_A(G) \in (0, 1]$, provided $Δ(G)$, $Δ_A(G)$, $Δ_B(G)$ are sufficiently large. These three bounds make progress towards three established conjectures: those of Erdős-Nešetřil (1985) for general graphs, Faudree-Gyárfás-Schelp-Tuza (1989) for bipartite graphs, and Brualdi-Quinn Massey (1993) in the asymmetric bipartite setting. Additionally, for the random bipartite graph $G \sim G(n_A, n_B, p)$ at constant $p \in (0,1)$ and bounded aspect ratio $\max(n_A, n_B) = O(\min(n_A, n_B))$, we prove the Brualdi-Quinn Massey bound $χ'_s(G) \leq Δ_A(G)\,Δ_B(G)$ asymptotically almost surely.

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