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 $Asubset mathcal{S}_0$ 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.
