On the sign patterns of entrywise positivity preservers in fixed dimension

Given a domain $I\subset\Bbb{C}$ and an integer $N>0$, function $f: I\to\Bbb{C}$ is said to be {\it entrywise positivity preserving} on positive semidefinite $N\times N$ matrices $A=(a_{jk})\in I^{N\times N}$, if the application $f[A]=(f(a_{jk}))$ of $f$ $A$ for all such $A$. Such preservers in dimensions have been classified by Schoenberg Rudin as being absolutely monotonic. In fixed dimension $N$, results akin work Horn Loewner show that first $N$ non-zero Maclaurin coefficients any preserver are positive; last also $I$ unbounded. However, very little was known about higher-order coefficients: only examples date unbounded domains were monotonic, hence dimensions; bounded non-absolutely monotonic few (and recent).