Let $S$ be a scheme and let $A$ be a commutative separated smooth $S$-group algebraic space of finite presentation each of whose geometric fibers is an extension of an abelian variety by a torus. Is $A$ necessarily a scheme?

If the geometric fibers are assumed to be abelian varieties (so that $A$ is necessarily proper), then the answer is positive, see Chai, Faltings "Degeneration of abelian varieties", Theorem 1.9 on page 5. The answer is positive also in the case when $S$ is a Dedekind scheme: this is proved in Anantharaman's thesis (respectively, already by M. Artin if $S$ is the spectrum of a field). Is the answer positive in general?

somesemi-abelian $S$-scheme with the same generic fiber. This one may hope to with moduli schemes, at least etale-locally on $S$. $\endgroup$ – user27920 Aug 10 '14 at 20:33