  Let $F$ be a CM field and $\Pi$ a regular algebraic cuspidal cohomological
representation of $\mathbf{G}=\operatorname{PGL}_2/F$. A conjecture of
Venkatesh describes the structure of the contribution of $\Pi$ to the homology
of the locally symmetric spaces associated to $\mathbf{G}$. We investigate this
conjecture in the setting of $p$-adic homology with $p$ a totally split prime.
Along the way, we elaborate on the relations between Venkatesh's conjecture and
completed homology, the Taylor-Wiles method and the $p$-adic local Langlands
correspondence. Our main result is a `big $R=T$' theorem in characteristic 0,
from which we deduce a variant of the $p$-adic realisation of Venkatesh's
conjecture, conditional on various natural conjectures and technical
assumptions.
