Three-space common ambient realization conjecture for Gromov-Hausdorff distance

Let $\mathcal{M}$ denote the collection of compact metric spaces. Given three compact metric spaces $X_1,X_2,X_3\in\mathcal{M}$, an isometric embedding $\varphi_i:X_i\hookrightarrow Z$ preserves all distances, and $d_\mathcal{H}^Z$ denotes Hausdorff distance in $Z$. Three-space common ambient realization conjecture. Given three compact metric spaces $X_1,X_2,X_3\in\mathcal{M}$, there exist $Z\in\mathcal{M}$ and isometric embeddings $\varphi_i:X_i\hookrightarrow Z$ such that

$$d_\mathcal{H}^Z\left(\varphi_i\left(X_i\right),\varphi_j\left(X_j\right)\right)=d_\mathcal{GH}\left(X_i,X_j\right)$$

for all $i,j\in\{1,2,3\}$. This is identified as the simplest unsolved case of the common-ambient-space question for finite collections of compact metric spaces not necessarily lying along a Gromov-Hausdorff geodesic.