Let $F$ be a totally real field of degree d. There are Hilbert modular varieties over $\mathbb{Q}$ that paramatrize abelian varieties of dimension d with an action of $\mathcal{O}_F$ the ring of integers of $F$ and some level structure.

Suppose we fix some level and we consider $\mathcal{A}\to M$, the universal abelian variety and for any positive integer $n$ the $n$-fold fiber product $\mathcal{A}^n\to M$ over $M$. This Kuga-Sato variety comes endowed with an action of $M_{n\times n}(\mathcal{O}_F)$. Do you know if these endomorphisms extend to the smooth toroidal compactifications denoted by $\overline{\mathcal{A}^n}$ in http://math.univ-lille1.fr/~mladen/articles/Pad16-DiTi.pdf ?

Thank you for your help!