  The enumeration of linear $\lambda$-terms has attracted quite some attention
recently, partly due to their link to combinatorial maps. Zeilberger and
Giorgetti (2015) gave a recursive bijection between planar linear normal
$\lambda$-terms and planar maps, which, when restricted to 2-connected
$\lambda$-terms (i.e., without closed sub-terms), leads to bridgeless planar
maps. Inspired by this restriction, Zeilberger and Reed (2019) conjectured that
3-connected planar linear normal $\lambda$-terms have the same counting formula
as bipartite planar maps. In this article, we settle this conjecture by giving
a direct bijection between these two families. Furthermore, using a similar
approach, we give a direct bijection between planar linear normal
$\lambda$-terms and planar maps, whose restriction to 2-connected
$\lambda$-terms leads to loopless planar maps. This bijection seems different
from that of Zeilberger and Giorgetti, even after taking the map dual. We also
explore enumerative consequences of our bijections.
