Let $A$ and $B$ be abelian schemes over a base scheme $S$. There is the $\underline{\mathrm{Hom}}(A,B)$ functor $T \mapsto \mathrm{Hom}(A \times T, B \times T)$, where $\mathrm{Hom}$ means homomorphisms of group schemes.

Is it true that $\underline{\mathrm{Hom}}(A,B)$ is representable by a scheme of locally finite presentation over $S$?

(By an abelian scheme over $S$, I mean a smooth proper group scheme over $S$, with geometrically integral fibers.)

I am ok with assuming that $S$ is locally Noetherian if necessary.

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If the abelian schemes $A$ and $B$ are moreover projective, one can make use of representability of the Hilbert scheme. I am not sure what to do if $A$ and $B$ are just proper, and not necessarily projective.

In the answer to this question

Representability of Hom of two finite flat group schemes

R. van Dobben de Bruyn mentions there is still an argument to show the representability of $\underline{\mathrm{Hom}}(A,B)$ by a scheme, but I have not yet found the argument.