Ideal rigidity conjecture for uncountable free Banach lattices

Let $\textswab{a}$ be an uncountable set, let $X$ be a projective Banach lattice, and suppose that $X$ contains a closed ideal isomorphic to the free Banach lattice $FBL(\textswab{a})$. Ideal rigidity conjecture. Under these hypotheses, one should have

$$X=FBL(\textswab{a}).$$

This is proposed as an improvement of an example showing that the Fremlin tensor product of $[?]$ and $FBL(\textswab{a})$ need not be projective when $\textswab{a}$ is uncountable; the supplied text gives no resolution.