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In this paper, we determine a constant occurring in a local analogue of theSiegel-Weil formula, and describe the behavior of the formal degrees under thelocal theta correspondence for quaternionic dual pairs of almost equal rankover a non-Archimedean local field of characteristic $0$. As an application, weprove the formal degree conjecture of Hiraga-Ichino-Ikeda for the non-splitinner forms of ${\rm Sp}_4$ and ${\rm GSp}_4$.

Formal degrees and the local theta correspondence: the quaternionic case