K-band GRAVITY/VLTI interferometry of "extreme" Herbig Be stars. The size-luminosity relation revisited

(Abridged:) It has been hypothesized that the location of Herbig Ae/Be stars (HAeBes) within the empirical relation between the inner disk radius (r$_{in}$), inferred from K-band interferometry, and the stellar luminosity (L$_*$), is related to the presence of the innermost gas, the disk-to-star accretion mechanism, the dust disk properties inferred from the spectral energy distributions (SEDs), or a combination of these effects. This work aims to test whether the previously proposed hypotheses do, in fact, serve as a general explanation for the distribution of HAeBes in the size-luminosity diagram. GRAVITY/VLTI spectro-interferometric observations at $\sim $2.2 $ \mu$m have been obtained for five HBes representing two extreme cases concerning the presence of innermost gas and accretion modes. V590 Mon, PDS 281, and HD 94509 show no excess in the near-ultraviolet, Balmer region of the spectra ($\Delta$D$_B$), indicative of a negligible amount of inner gas and disk-to-star accretion, whereas DG Cir and HD 141926 show such strong $\Delta$D$_B$ values that cannot be reproduced from magnetospheric accretion, but probably come from the alternative boundary layer mechanism. Additional data for these and all HAeBes resolved through K-band interferometry have been compiled from the literature and updated using Gaia EDR3 distances, almost doubling previous samples used to analyze the size-luminosity relation. We find no general trend linking the presence of gas inside the dust destruction radius or the accretion mechanism with the location of HAeBes in the size-luminosity diagram. Underlying trends are present and must be taken into account when interpreting the size-luminosity correlation. Still, it is argued that the size-luminosity correlation is most likely to be physically relevant in spite of the previous statistical warning concerning dependencies on distance.