  We apply the theory of fundamental strata of Bremer and Sage to find
cohomologically rigid $G$-connections on the projective line, generalising the
work of Frenkel and Gross. In this theory, one studies the leading term of a
formal connection with respect to the Moy-Prasad filtration associated to a
point in the Bruhat-Tits building. If the leading term is regular semisimple
with centraliser a (not necessarily split) maximal torus $S$, then we have an
$S$-toral connection. In this language, the irregular singularity of the
Frenkel-Gross connection gives rise to the homogenous toral connection of
minimal slope associated to the Coxeter torus $\mathcal{C}$. In the present
paper, we consider connections on $\mathbb{G}_m$ which have an irregular
homogeneous $\mathcal{C}$-toral singularity at zero of slope $i/h$, where $h$
is the Coxeter number and $i$ is a positive integer coprime to $h$, and a
regular singularity at infinity with unipotent monodromy. Our main result is
the characterisation of all such connections which are rigid.
