Weak units and homotopy 3-types

We show that every braided monoidal category arises as $\End(I)$ for a weak unit $I$ in an otherwise completely strict monoidal 2-category. This implies a version of Simpson's weak-unit conjecture in dimension 3, namely that one-object 3-groupoids that are strict in all respects, except that the object has only weak identity arrows, can model all connected, simply connected homotopy 3-types. The proof has a clear intuitive content and relies on a geometrical argument with string diagrams and configuration spaces.