New results are reported from tests of a low-energy effective field theory (EFT) that includes a dilaton field to describe the emergent light scalar with ${ 0^{++} }$ quantum numbers in the strongly coupled near-conformal gauge theory with a massless fermion flavor doublet in the two-index symmetric (sextet) representation of the SU(3) color gauge group. In the parlor of walking --- based on the observed light scalar, the small $\beta$-function at strong coupling, and the large anomalous scale dimension of the chiral condensate --- the dilaton EFT hypothesis is introduced to test if it explains the slowly changing nearly scale invariant physics that connects the asymptotically free UV fixed point and the far-infrared scale of chiral symmetry breaking. The characteristic dilaton EFT signatures of scale symmetry breaking are probed in this report in the small Compton wavelength limit of Goldstone bosons relative to the size of the lattice volume (p-regime) and in the limit when the Goldstone wavelength exceeds the size of the volume ($\epsilon$-regime). Random matrix theory (RMT) analysis of the dilaton EFT is applied to the lowest part of the Dirac spectrum in the $\epsilon$-regime to directly test predictions for the fundamental EFT parameters. The predictions, sensitive to the choice of the dilaton potential, were limited before to the p-regime, using extrapolations from far above the chiral limit with untested uncertainties. The dilaton EFT analysis of the $\epsilon$-regime was first suggested in~\cite{Fodor:2019vmw}, with some results presented at this conference and with our continued post-conference analysis added to stimulate discussions.