Partitioning the double $\beta$ decay of $^{116}$Cd: Electron-capture on $^{116}$In

$^{116}$Cd undergoes double $\beta$ decay to $^{116}$Sn. This rare process may be described by virtual transitions through states in the intermediate nucleus $^{116}$In and there could be a major contribution from the $^{116}$In ground state. It is, therefore, valuable to determine the matrix elements for the decays of $^{116}$In to $^{116}$Cd and $^{116}$Sn in order to provide benchmarks for nuclear models that are used to calculate double $\beta$ decay matrix elements. We have experimentally constrained the small decay branch for electron capture on the ground state of $^{116}$In. Samples of $^{116}$In were produced using the IGISOL facility at the University of Jyv{\"a}skyl{\"a}, isobarically purified using the JYFLTRAP double-Penning trap, and delivered to a counting station consisting of a high-purity Ge x-ray detector and a 4$\pi$ scintillator to discriminate $\beta$-decay events. Our result sheds light on discrepancies between past values for the Gamow-Teller strength obtained via electron capture and charge-exchange, providing a more consistent benchmark in the $A=116$ system for nuclear models of double $\beta$ decay.