  The aim of this paper is to show that almost greedy bases induce tighter
embeddings in superreflexive Banach spaces than in general Banach spaces. More
specifically, we show that an almost greedy basis in a superreflexive Banach
space $\mathbb{X}$ induces embeddings that allow squeezing $\mathbb{X}$ between
two superreflexive Lorentz sequence spaces that are close to each other in the
sense that they have the same fundamental function.
