Lipschitz magnitude conjecture for tractable subspaces

Let $L_1$ be the space of integrable real-valued functions with its usual metric, and let $T\subseteq L_1$ be a tractable subspace. Write $\mathrm{Fin}_{+}(T)$ and $\mathrm{Cmp}_{+}(T)$ for the spaces of finite and compact subspaces of $T$, respectively, and let $\mathsf{mag}_{\mathrm{Fin}_{+}(T)}$ and $\mathsf{mag}_{\mathrm{Cmp}_{+}(T)}$ denote their magnitude maps. A map is Lipschitz if there is a constant $C\geq 0$ such that its output differences are at most $C$ times the distances between inputs.

Lipschitz magnitude conjecture. Let $T\subseteq L_1$ be tractable. Then

$$\mathsf{mag}_{\mathrm{Cmp}_{+}(T)}\colon \mathrm{Cmp}_{+}(T)\to\mathbb{R}$$

is Lipschitz if and only if $T$ is bounded. Equivalently, the magnitude map on $\mathrm{Fin}_{+}(T)$ is Lipschitz if and only if $T$ is bounded; for tractable $T$, boundedness is equivalent to compactness.

This conjecture characterizes exactly when magnitude has global Lipschitz continuity on compact or finite subspaces of a tractable subspace of $L_1$.