Let's work over a trait $S=\mathrm{Spec}R$, where $R$ is a dvr with fraction field $K$, residue field $k$. Given an abelian variety $A_K$ with semi-stable reduction, let $A$ over $S$ be its Néron model and $A^{\circ}$ the neutral component. We know the sheaf $\mathscr{E}xt^1(A^{\circ},\mathbb{G}_m)$ is represented by the Néron model of the dual of $A_K$ over the category of smooth scheme over $S$, see (Mazur and Messing's LNM Universal extensions and one dimensional crystalline cohomology, chapter I section 5)

My question is: Is the sheaf $\mathscr{E}xt^1(A^{\circ},\mathbb{G}_m)$ also representable over the category of schemes over $S$?

Also we know the Poincaré biextension $W_K$ of $A_K$ and $A_K^'$ by $\mathbb{G}_m$ extends to a biextension $W$ of certain open subgroups of $A$ and $A'$ (i.e. the subgroups making the component pairing vanish), my second question is if $W$ is represented by a scheme over $S$?