  We prove in this paper the original version of Kontsevich and Soibelman's
motivic integral identity conjecture for formal functions by developing a novel
framework for equivariant motivic integration on special rigid varieties. This
theory is built upon our recent research on equivariant motivic integration
within the realm of special formal schemes. The central element of our approach
lies in demonstrating that two formal models of a given smooth rigid variety
can be dominated by a third formal model. Notably, a similar assertion for
quasi-compact rigid varieties was obtained by Bosch, L\"utkebohmert, and
Raynaud in 1993. Consequently, we establish a concept of motivic volume for a
special smooth rigid variety, ensuring independence from the selection of its
models. We demonstrate that this motivic volume can be extended to a
homomorphism from a certain Grothendieck ring of special smooth rigid varieties
to the classical Grothendieck ring of varieties. Moreover, our developed
motivic volume exhibits a Fubini-type property, which recovers Nicaise and
Payne's motivic Fubini theorem for the tropicalization map.
