Dilatorial characterization conjecture for beta-models of strong systems

Let $T$ be a strong system with an ordinal analysis based on ordinal representation systems $\mathsf{OT}$. These systems give rise to functors

$$F_{\!_{\mathsf{OT}}}:\mathsf{DIL}\longrightarrow\mathsf{WO}$$

that send dilators to ordinal representation systems. Dilatorial characterization conjecture. The assertion that $F_{\!_{\mathsf{OT}}}$ sends dilators to well-orderings is equivalent to the statement that every set is contained in a countably coded $\beta$-model of $T$. The claim is presented as a general conjecture extending the established characterization for $\beta$-models of $\Pi^1_1$-comprehension, but no proof or resolution is supplied.