  Let $k$ be a non-Archimedean rational valued field. We construct the moduli
space of linearly rigidified polarized analytic tori over $k$ that admit
rigid-analytic uniformization by an algebraic torus and observe that it is in
definable rigid subanalytic bijection with a $PGL_N$-bundle over a polyhedral
domain in an algebraic torus. We use this observation to prove that the
Hrushovski-Kazhdan motivic volume of a non-Archimedean semi-algebraic family of
Abelian varieties admitting such a uniformization fibrewise vanishes. This
question is motivated by the conjectural geometric interpretation of tropical
refined multiplicities of Block and Goetsche proposed by Nicaise, Payne and
Schroeter.
