Freeness conjecture for projective Banach lattices without a topological order unit

Let $X$ be a projective Banach lattice, and let a topological order unit mean an element whose principal order ideal is dense in $X$. A Banach lattice is free when it is a free Banach lattice over some set of generators. Freeness conjecture. If $X$ does not have a topological order unit, then $X$ must be free. The conjecture is presented as a possible structural description of projective Banach lattices beyond the separable, topologically order-unital examples, and no resolution is given in the supplied text.