  We elaborate further on the compatibility of the "vacuumon potential" that
characterises the inflationary phase of the Running Vacuum Model (RVM) with the
Swampland criteria. The work is motivated by the fact that, as demonstrated
recently by the authors [1-3], the RVM framework can be derived as an effective
gravitational field theory stemming from underlying microscopic (critical)
string theory models with gravitational anomalies, involving condensation of
primordial gravitational waves. Although believed to be a classical scalar
field description, not representing a fully fledged quantum field, nonetheless
we show here that the vacuumon potential satisfies certain Swampland criteria
for the relevant regime of parameters and field range. We link the criteria to
the Gibbons-Hawking entropy that has been argued to characterise the RVM during
the de Sitter phase. These results imply that the vacuumon may, after all,
admit under certain conditions, a r\^ole as a quantum field during the
inflationary (almost de Sitter) phase of the running vacuum. The conventional
slow-roll interpretation of this field, however, fails just because it
satisfies the Swampland criteria. The RVM effective theory derived from the
low-energy effective action of string theory does, however, successfully
describe inflation thanks to the $\sim H^4$ terms induced by the gravitational
anomalous condensates. In addition, the stringy version of the RVM involves the
Kalb-Ramond (KR) axion field, which, in contrast to the vacuumon, does
perfectly satisfy the slow-roll condition. We conclude that the vacuumon
description is not fully equivalent to the stringy formulation of the RVM.
