  Starting with a vertex-weighted pointed graph $(\Gamma,\mu,v_0)$, we form the
free loop algebra $\mathcal{S}_0$ defined in Hartglass-Penneys' article on
canonical $\rm C^*$-algebras associated to a planar algebra. Under mild
conditions, $\mathcal{S}_0$ is a non-nuclear simple $\rm C^*$-algebra with
unique tracial state. There is a canonical polynomial subalgebra $A\subset
\mathcal{S}_0$ together with a Dirac number operator $N$ such that $(A,
L^2A,N)$ is a spectral triple. We prove the Haagerup-type bound of
Ozawa-Rieffel to verify $(\mathcal{S}_0, A, N)$ yields a compact quantum metric
space in the sense of Rieffel.
  We give a weighted analog of Benjamini-Schramm convergence for
vertex-weighted pointed graphs. As our $\rm C^*$-algebras are non-nuclear, we
adjust the Lip-norm coming from $N$ to utilize the finite dimensional
filtration of $A$. We then prove that convergence of vertex-weighted pointed
graphs leads to quantum Gromov-Hausdorff convergence of the associated adjusted
compact quantum metric spaces.
  As an application, we apply our construction to the
Guionnet-Jones-Shyakhtenko (GJS) $\rm C^*$-algebra associated to a planar
algebra. We conclude that the compact quantum metric spaces coming from the GJS
$\rm C^*$-algebras of many infinite families of planar algebras converge in
quantum Gromov-Hausdorff distance.
