Scaling limit conjecture for first-passage percolation geodesics on planar maps

Let $a\from (3/2,5/2)$, let $\bigl(f_i\bigr)_{i\geq 1}$ be the sampled marked vertices or faces, let $d^\dagger_\mathrm{fpp}$ denote the first-passage percolation distance, and let $\bigl(w_i\bigr)_{i\geq 1}$ be the corresponding points of the continuum metric space $(\mathcal{D}_a,d_{\mathcal{D}_a})$. Here $c_a$ and $p_{\bf q}$ are the constants appearing in the model, and convergence in distribution is taken in the product topology. First-passage percolation scaling-limit conjecture. Under $\mathbb{P}^{(\ell)}$,

$$\left(\ell^{2-a}d^\dagger_\mathrm{fpp}(f_i,f_j)\right)_{i,j\geq 1}\mathop{\longrightarrow}\limits_{\ell\to\infty}^{(\mathrm{d})}\left(\frac{1}{2c_ap_{\bf q}}d_{\mathcal{D}_a}(w_i,w_j)\right)_{i,j\geq 1}.$$

This conjecture identifies the scaling limit of the first-passage percolation metric with the metric on the continuum space coded by the limiting tree and shortcuts. The supplied text does not state whether the conjecture has been proved or disproved.