  We reinterpret a conjecture of Breuil on the locally analytic
$\mathrm{Ext}^1$ in a functorial way using $(\varphi,\Gamma)$-modules (possibly
with $t$-torsion) over the Robba ring, making it more accurate. Then we prove
several special or partial cases of this "improved" conjecture, notably for
${\rm GL}_3(\mathbb{Q}_p)$.
