The converse generation conjecture for order-invariant measures

Let $P=(Z,<)$ be a causal set, let $\Omega$ be its space of natural extensions, and let $\mu$ be an order-invariant measure on the associated measurable space. An element $\omega=x_1x_2\cdots\in\Omega$ generates $\mu$ if $\nu^k(\mathord\cdot)(\omega)$ converges weakly to $\mu$ as $k\to\infty$. Generation conjecture. If $\mu$ is an order-invariant measure that is generated by some $\omega\in\Omega$, then $\mu$ is extremal. Extremal order-invariant measures are known to be generated by $\mu$-almost every element of $\Omega$; the conjecture asks whether generation by even one element forces extremality, providing the converse to the established almost-sure implication.