Magnitude–intrinsic-volume finiteness conjecture for compact convex subsets of $L_1$

Let $L_1$ denote the space of integrable functions, and let $K\subseteq L_1$ be compact and convex. Write $\mu_1^{L_1}(K)$ for the supremum of the first Holmes–Thompson intrinsic volume over finite-dimensional affine sections of $K$. Magnitude–intrinsic-volume finiteness conjecture.

$$\operatorname{Mag}\left(K,\left\Vert\cdot\right\Vert_1\right)<\infty\quad\Longleftrightarrow\quad \mu_1^{L_1}(K)<\infty.$$

The conjecture seeks a converse to the known implication supplied by the paper’s $L_1$ one-point result: finite intrinsic-volume control implies finite magnitude. The converse is presented as open and is motivated by earlier Hilbert-space results and examples of compact positive-definite metric spaces with infinite magnitude.