  Suppose that $\Gamma$ is a conformal loop ensemble (CLE$_\kappa$) with simple
loops ($\kappa \in (8/3,4)$) in a simply connected domain $D \subseteq {\mathbf
C}$ whose boundary is itself a type of CLE$_\kappa$ loop. Let $\Upsilon$ be the
carpet of $\Gamma$, i.e., the set of points in $D$ not surrounded by a loop of
$\Gamma$. We prove that certain approximations to the chemical distance metric
in $\Upsilon$ are tight. More precisely, for each path $\omega \colon [0,1] \to
\Upsilon$ and $\epsilon > 0$ we let ${\mathfrak N}_\epsilon(\omega)$ be the
Lebesgue measure of the $\epsilon$-neighborhood of $\omega$. For $z,w \in
\Upsilon$ we let ${\mathfrak d}_\epsilon(z,w;\Gamma) = \inf_\omega {\mathfrak
N}_\epsilon(\omega)$ where the infimum is over all paths $\omega \colon [0,1]
\to \Upsilon$ with $\omega(0) = z$, $\omega(1) = w$ and let ${\mathfrak
m}_\epsilon$ be the median of $\sup_{z,w \in \partial D} {\mathfrak
d}_\epsilon(z,w;\Gamma)$. We prove that $(z,w) \mapsto {\mathfrak
m}_\epsilon^{-1} {\mathfrak d}_\epsilon(z,w;\Gamma)$ is tight and that any
subsequential limit defines a geodesic metric on $\Upsilon$ which is H\"older
continuous with respect to the Euclidean metric. We conjecture that the
subsequential limit is unique, conformally covariant, and describes the scaling
limit of the chemical distance metric for discrete loop models which converge
to CLE$_\kappa$ for $\kappa \in (8/3,4)$ such as the critical Ising model.
