Random band matrices are a natural model for studying the quantum propagation in disordered systems as they interpolate between random Schrödinger operators and Wigner matrices. We study the random band matrix ensemble introduced by Disertori--Pinson--Spencer, characterised by the variance profile associated with the operator $(-W^{2}\Delta+1)^{-1}$, $\Delta$ being the Laplacian on the three-dimensional lattice and $W$ being the characteristic width of the band. For any energy $E$ close to the spectral edge, namely for $1.8 <|E| <2$, we rigorously prove that the density of states follows Wigner semicircle law with precision $W^{-2}$, provided $W$ is sufficiently large, depending on $E$. The proof relies on the supersymmetric approach of Disertori--Pinson--Spencer, and extends their result, which was previously established for energies $|E| \leq 1.8$. Joint work with M. Disertori.