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.
