Sign regularity preserving linear operators

A matrix $A\in {\mathbb{R}}^{m\times n}$ is strictly sign regular/SSR (or sign regular/SR) if for each $1\le k\le \min \{m,n\}$, all (non-zero) $k\times k$ minors of $A$ have the same sign. This class of matrices contains the totally positive matrices, and was first studied by Schoenberg in 1930 to characterize variation diminution, a fundamental property in total positivity theory. In this article, we classify all surjective linear mappings $\mathcal{L}:{\mathbb{R}}^{m\times n}\to {\mathbb{R}}^{m\times n}$ that preserve: (i) sign regularity and (ii) sign regularity with a given sign pattern, as well as (iii) strict versions of these.