The enrichment–module equivalence for symmetric monoidal Lawvere theories

Let $\mathcal{L}$ be a Lawvere theory admitting a symmetric monoidal structure compatible with finite products. Consider the $\infty$-categories of $\operatorname{Mdl}_{\mathcal{L}}$-enriched Lawvere theories and $\mathcal{L}$-module Lawvere theories. The enrichment–module conjecture. These $\infty$-categories are equivalent. This would provide a converse to the result that an $\mathcal{L}$-compatible symmetric monoidal structure induces enrichment of modules over $\mathcal{L}$; the paper presents it as a suggested strong converse, and no proof or resolution is given.