Néron mapping property for logarithmic abelian varieties

Let $A\to S$ be a family of log abelian varieties over a log smooth base $S$. Let $U\subseteq S$ be the dense open subset of $S$ on which the log structure is trivial. For a log smooth morphism $X\to S$, the Néron mapping property asserts that the restriction map

$$\operatorname{\textup{Hom}}_S(X,A)\to \operatorname{\textup{Hom}}_U(X|_U,A|_U)$$

is an isomorphism. Néron mapping property conjecture. Every family of log abelian varieties $A\to S$ over a log smooth base $S$ satisfies the Néron mapping property for log smooth morphisms. The property is known for logarithmic Jacobians and logarithmic tori, but is not proven in the literature for more general log abelian varieties; it is expected to hold.