We investigate the behavior of the dense and cold (normalized) quantum chromodynamics (QCD) trace anomaly, Δ, in the interior of twin neutron stars (obtained from several sets of equations of state in agreement with modern compact-star and multimessenger data) satisfying static and dynamic stability conditions. We scan the formed twin-star parameter space in order to look for effects caused by the presence of a strong first-order phase transition connecting hadron and quark phases by means of a Maxwell construction. We found robustly that Δ suffers an abrupt decrease around the transition point, even reaching large negative values (Δ ≃ −0.35), in marked contrast to current studies pointing out a smooth behavior with Δ ≳ 0 at all densities. Besides, we characterize the behavior of the recently defined conformal factor, dc, in the four categories of twin stars for which we perfor...