  On general Carnot groups, the definition of a possible hypoelliptic
Hodge-Laplacian on forms using the Rumin complex has been considered by Rumin,
who introduced a 0-order pseudodifferential operator on forms. However, for
questions regarding regularity for example, where one needs sharp estimates,
this 0-order operator is not suitable. Up to now, there have only been very few
attempts to define hypoelliptic Hodge-Laplacians on forms that would allow for
such sharp estimates. Indeed, this question is rather difficult to address in
full generality, the main issue being that the Rumin exterior differential
$d_c$ is not homogeneous on arbitrary Carnot groups.
  In this note, we consider the specific example of the free Carnot group of
step 3 with 2 generators, and we introduce three possible definitions of
hypoelliptic Hodge-Laplacians. We compare how these three possible Laplacians
can be used to obtain sharp div-curl type inequalities akin to those considered
by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their
subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin
complex on Heisenberg groups.
