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[en] We analyze quantum no-scale regimes (QNSR) in perturbative heterotic string compactified on tori, with total spontaneous breaking of supersymmetry. We show that for marginal deformations initially at any point in moduli space, the dynamics of a flat, homogeneous and isotropic universe can always be attracted to a QNSR. This happens independently of the characteristics of the 1-loop effective potential , which can be initially positive, negative or vanishing, and maximal, minimal or at a saddle point. In all cases, the classical no-scale structure is restored at the quantum level, during the cosmological evolution. This is shown analytically by considering moduli evolutions entirely in the vicinity of their initial values. Global attractor mechanisms are analyzed numerically and depend drastically on the sign of . We find that all initially expanding cosmological evolutions along which is positive are attracted to the QNSR describing a flat, ever-expanding universe. On the contrary, when can reach negative values, the expansion comes to a halt and the universe eventually collapses into a Big Crunch, unless the initial conditions are tuned in a tiny region of the phase space. This suggests that flat, ever-expanding universes with positive potentials are way more natural than their counterparts with negative potentials.