Arzela–Ascoli conjecture with addresses for Gromov–Hausdorff converging spaces

Let $(X_j,d_{X_j})$ and $(Y_j,d_{Y_j})$ be sequences of compact metric spaces such that

$$X_j \stackrel{\textrm{GH}}{\longrightarrow} X_\infty \qquad\textrm{and}\qquad Y_j \stackrel{\textrm{GH}}{\longrightarrow} Y_\infty.$$

Suppose that $F_j:X_j\to Y_j$ are uniformly bounded $K$-Lipschitz functions for some $K>0$. Arzela–Ascoli conjecture with addresses. A subsequence converges to a bounded $K$-Lipschitz function

$$F_\infty:X_\infty\to Y_\infty.$$

This is proposed as a stronger version of the preceding Arzela–Ascoli theorem with addresses, extending the conclusion from real-valued functions on varying spaces to functions whose domains and targets both vary in the Gromov–Hausdorff sense. The source does not provide a resolution of this stronger statement.