Seeking a coherent explanation of LHC excesses for compressed spectra

The most recent searches by the ATLAS and CMS Collaborations in final states with soft leptons and missing transverse energy show mild excesses predominantly associated with dilepton invariant masses of about 10-20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10{-}20$$\end{document} GeV, which can result from decays of electroweakinos that are heavier than the lightest neutralino by O(10)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(10)$$\end{document} GeV. On the other hand, these analyses are insensitive to electroweakino mass splittings smaller than about 5 GeV. In previous work, we demonstrated that while recent searches in the monojet channel can exclude some of the smallest O(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(1)$$\end{document} GeV mass splitting configurations for electroweakinos, they also exhibit excesses that can overlap with the soft-lepton excesses in certain models, including a simplified scenario with pure higgsinos. In this work we dive deeper into these excesses, studying the analyses in detail and exploring an array of models that go beyond the simplified scenarios considered by the experimental collaborations. We show that, in the Minimal Supersymmetric Standard Model, the overlapping excesses are not unique to the pure-higgsino limit, instead persisting in realistic parameter space featuring a bino-like lightest supersymmetric particle with some wino admixture. On the other hand, for the Next-to-Minimal Supersymmetric Standard Model with a singlino-like lightest supersymmetric particle and higgsino-like next-to-lightest supersymmetric particle(s), the excess in the two-lepton channel fits rather well with the parameter space predicting the correct relic abundance through freeze out, but the monojet fit is much poorer. Interestingly, the excesses either do not overlap or do not exist at all for two non-supersymmetric models seemingly capable of producing the correct final states.