The implication from a a6-set theoretic basis to the positive basis property

Let $H$ be a Hopf $k$-algebra with ${\rm char}(k)=0$.

Expected implication. If $H$ has a $\Phi$-set theoretic basis, then $H$ has the positive basis property.

The authors expect this implication in general, while the converse is not expected: they note that nearly positive Hopf algebras such as the universal enveloping algebra $U(\mathfrak{g})$ of a finite dimensional Lie algebra can lack a $\Phi$-set theoretic basis. The implication is established in the finite dimensional setting by the preceding classification result, but remains conjectural for arbitrary Hopf $k$-algebras.