Some finite dimensional problems of phase retrieval in Banach lattices

In the Banach lattice setting, Phase Retrieval consists of recovering $f$ (up to a sign) from its modulus $|f|$, if certain additional information about $f$ is known; usually, $f$ is assumed to belong to a certain given subspace of our lattice. Stable Phase Retrieval (SPR) requires that $f$ be reconstructed “in a robust manner”. We address several questions concerning performing SPR for finite dimensional subspaces.
(i) Suppose we are given a finite dimensional subspace $F$ of a Banach lattice $X$. Does $F$ have SPR subspaces, and if yes, what is their dimension?
(ii) Conversely, suppose we are given a finite dimensional Banach space $E$. What is the smallest possible Banach lattice $X$ into which $E$ can be embedded in an SPR way?
(Joint work with D. Freeman, B. Pineau, M. Taylor)