Separability conjecture for projective Banach lattices with a topological order unit

A projective Banach lattice is a Banach lattice with the projective lifting property, and a topological order unit is an element whose principal order ideal is dense in the lattice. Separability conjecture. If a projective Banach lattice has a topological order unit, then it is separable. This is proposed as a possibly rash conjecture because the known building blocks for projective Banach lattices in the surrounding discussion are separable and therefore have topological order units; its status is not resolved in the supplied text.