In-beam $\gamma$-ray spectroscopy towards the proton dripline: The curious case of $^{32}$Ar

High-resolution in-beam $\gamma$-ray spectroscopy was used to study excited states of the neutron-deficient nucleus $^{32}$Ar populated in fast-beam induced four- and six-nucleon removal reactions from $^{36,38}$Ca. One new $\gamma$-ray transition and indications for an additional two were found, allowing for a glimpse at the level scheme beyond the known $2^+_1$ state. The nature of the new $1900(4)$-keV transition is discussed in the context of the known energy spectrum of the mirror nucleus $^{32}$Si and shell-model calculations using the FSU and SDPF-M cross-shell effective interactions. Its resulting parent state at $3767(5)$ keV, more than $1.3$ MeV above the proton separation energy, is tentatively assigned to have mixed sd-shell and $2p$-$2h$ character. It might either be the mirror of the $J^{\pi}=2^+_2$ state of $^{32}$Si at $4230.8(8)$ keV, but with a decay branch favoring a transition to the $2^+_1$ over the ground state, or the mirror of the $4983.9(11)$-keV state with quantum numbers $0^+$. The resulting mirror-energy differences of $-473(5)$ and $-1218(5)$ keV are both sizable when compared to systematics; in the latter case it would, in fact, be among the largest reported to date in the entire nuclear chart or suggest the potential existence of an additional, hitherto unidentified, low-lying $0^+$ state of $^{32}$Si.