  Building upon recent results of Dub\'edat (see arXiv:1403.6076) on the
convergence of topological correlators in the double-dimer model considered on
Temperleyan approximations $\Omega^\delta$ to a simply connected domain
$\Omega\subset\mathbb C$ we prove the convergence of probabilities of
cylindrical events for the \emph{double-dimer loop ensembles} on
$\Omega^\delta$ as $\delta\to 0$. More precisely, let
$\lambda_1,\dots,\lambda_n\in\Omega$ and $L$ be a macroscopic lamination on
$\Omega\setminus\{\lambda_1,\dots,\lambda_n\}$, i.e., a collection of disjoint
simple loops surrounding at least two punctures considered up to homotopies. We
show that the probabilities $P_L^\delta$ that one obtains $L$ after withdrawing
all loops surrounding no more than one puncture from a double-dimer loop
ensemble on $\Omega^\delta$ converge to a conformally invariant limit $P_L$ as
$\delta \to 0$, for each $L$.
  Though our primary motivation comes from 2D statistical mechanics and
probability, the proofs are of a purely analytic nature. The key techniques are
the analysis of entire functions on the representation variety
$\mathrm{Hom}(\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb
C))$ and on its (non-smooth) subvariety of locally unipotent representations.
In particular, we do \emph{not} use any RSW-type arguments for double-dimers.
  The limits $P_L$ of the probabilities $P_L^\delta$ are defined as
coefficients of the isomonodormic tau-function studied by Dub\'edat with
respect to the Fock--Goncharov lamination basis on the representation variety.
The fact that $P_L$ coincides with the probability to obtain $L$ from a sample
of the nested CLE(4) in $\Omega$ requires a small additional input, namely a
mild crossing estimate for this nested conformal loop ensemble.
