Regularizability conjecture for bimeromorphic maps of polarized abelian families

Let $f\colon \mathcal{X}\dashrightarrow \mathcal{X}$ be a bimeromorphic map of a family of polarized abelian varieties that fixes the $0$-section. A regularizability conjecture asserts that $f$ is regularizable if and only if, possibly after base change, there is an $f$-invariant splitting

$$\mathcal{X}=\mathcal{Y}\times\mathcal{Y}'$$

where $\mathcal{Y}$ is a non-degenerating family of polarized automorphisms of abelian varieties, while $\mathcal{Y}'$ is a family of polarized abelian varieties and $f|_{\mathcal{Y}'}$ is a family of automorphisms such that

$$f^*_t|_{\mathcal{Y}'}\colon H^1(Y'_t,\mathbb{Z})\to H^1(Y'_t,\mathbb{Z})$$

has finite order. The conjecture proposes a structural characterization of regularizable maps: after base change, the non-degenerating and finite-order parts should split off from the family.