We prove an almost optimal lower bound for the so called last ‘geometric’ minimum for the height of a subvariety V of a power of the multiplicative group ${\mathbb G}_m^n$. More precisely, one shows that if $\boldsymbol x$ is a point of $V(\overline{\mathbb Q})$ which does not belong to a translate of a sub-torus of ${\mathbb G}_m^n$ lying in V, of height less than a function essentially linear in the inverse of the degree of V, then $\boldsymbol x$ belongs to a finite ‘exceptional’ subset of V. Thus, one proves ‘up to an $\varepsilon$’ the sharpest conjectures that can be formulated on this problem. Previously, the best known result in this direction was due to the second author and Philippon, who provided lower bounds which were inverse monomial in the degree of V (Minorations des hauteurs normalisées des sous-variétés des tores, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 489–543; Errata29 (2000)).