Formal degrees and the local theta correspondence: the quaternionic case

In this paper, we determine a constant occurring in a local analogue of the Siegel-Weil formula, and describe the behavior of the formal degrees under the local theta correspondence for quaternionic dual pairs of almost equal rank over a non-Archimedean local field of characteristic $0$. As an application, we prove the formal degree conjecture of Hiraga-Ichino-Ikeda for the non-split inner forms of ${\mathrm{S}\mathrm{p}}_4$ and ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$.