Characterisation of minimal hereditary graph classes of unbounded clique-width

Let $\mathcal{G}^\delta$ be the hereditary graph class associated with the sequence $\delta$, and let $\Delta_{\min}$ denote the set of sequences satisfying the defining minimality conditions, including recurrence, bounded clique-width for gap factors, and bounded $\mathcal{M}^\beta$ for the bond set $\beta$.

Characterisation conjecture. The hereditary graph class $\mathcal{G}^\delta$ is minimal of unbounded clique-width if and only if $\delta \in \Delta_{\min}$.

The sufficient conditions for membership in $\Delta_{\min}$ are known to yield minimal hereditary classes of unbounded clique-width, and recurrence together with bounded clique-width for gap factors is necessary. The unresolved part is whether the bond set must also have bounded $\mathcal{M}^\beta$; no counterexample with $\delta \notin \Delta_{\min}$ is known.