Free bianalytic maps between spectrahedra and spectraballs in a generic setting

Given a tuple $E=(E_1,\dots ,E_g)$ of $d\times d$ matrices, the collection of those tuples of matrices $X=(X_1,\dots ,X_g)$ (of the same size) such that $| \sum E_j\otimes X_j|\le 1$iscalledaspectraball$\mathcal B_E$. Likewise, given a tuple $B=(B_1,\dots ,B_g)$ of $e\times e$ matrices the collection of tuples of matrices $X=(X_1,\dots ,X_g)$ (of the same size) such that $I + \sum B_j\otimes X_j +\sum B_j^* \otimes X_j^*\succeq 0$isafreespectrahedron$\mathcal D_B$\mathrm{.}Assuming$E$and$B$ are irreducible, plus an additional mild hypothesis, there is a free bianalytic map $p:{\mathcal{B}}_E\to {\mathcal{D}}_B$ normalized by $p(0)=0$ and $p^{\mathrm{\prime }}(0)=I$ if and only if ${\mathcal{B}}_E={\mathcal{B}}_B$ and $B$ spans an algebra. Moreover $p$ is unique, rational and has an elegant algebraic representation.