We consider the Glauber dynamics for the 2D Ising model in a box of side L, at inverse temperature β and random boundary conditions τ whose distribution P either stochastically dominates the extremal plus phase (hence the quotation marks in the title) or is stochastically dominated by the extremal minus phase. A particular case is when P is concentrated on the homogeneous configuration identically equal to + (equal to −). For β large enough we show that for any ε > 0 there exists c = c(β, ε) such that the corresponding mixing time Tmix satisfies limL→∞ P (Tmix ≥ exp(cLε)) = 0. In the non-random case τ ≡ + (or τ ≡ −), this implies that Tmix ≤ exp(cLε). The same bound holds when the boundary conditions are all + on three sides and all − on the remaining one. The result, although still very far from the expected Lifshitz behavior Tmix = O(L2), considerably improves upon the previous known estimates of the form Tmix ≤ exp(cL 12 +ε). The techniques are based on induction over length scales, combined with a judicious use of the so-called “censoring inequality” of Y. Peres and P. Winkler, which in a sense allows us to guide the dynamics to its equilibrium measure.