Generalized $T_e$([OIII])-$T_e$(HeI) Discrepancies in Ionized Nebulae: Possible Evidence of Case B Deviations and Temperature Inhomogeneities

The physics of recombination lines (RLs) in the HeI singlet system is expected to be relatively simple, supported by accurate atomic models. We examine the intensities of HeI singlets $\lambda \lambda$3614, 3965, 5016, 6678, 7281 and the triplet HeI $\lambda$5876 in various types of ionized nebulae and compare them with theoretical predictions to test the validity of the ``Case B'' recombination scenario and the assumption of thermal homogeneity. Our analysis includes 85 spectra from Galactic and extragalactic HII regions, 90 from star-forming galaxies, and 218 planetary nebulae, all compiled by the DEep Spectra of Ionized REgions Database Extended (DESIRED-E) project. By evaluating the ratios HeI $\lambda$7281/$\lambda$6678 and HeI $\lambda$7281/$\lambda$5876, we determine $T_e$(HeI) and compare it with direct measurements of $T_e$([OIII] $\lambda$4363/$\lambda$5007). We find that $T_e$(HeI) is systematically lower than $T_e$([OIII]) across most objects and nebula types. Additionally, we identify a correlation between the abundance discrepancy factor (ADF(O$^{2+}$)) and the difference $T_e$([OIII]) - $T_e$(HeI) for planetary nebulae. We explore two potential explanations: photon loss from $n^1P \rightarrow 1^1S$ transitions and temperature inhomogeneities. Deviations from ``Case B'' may indicate photon absorption by HI rather than HeI and/or generalized ionizing photon escape, highlighting the need for detailed consideration of radiative transfer effects. If temperature inhomogeneities are widespread, identifying a common physical phenomenon affecting all ionized nebulae is crucial. Our results suggest that both scenarios can contribute to the observed discrepancies.