Extension of an independence relation to the lovely-pair structure

Let $\mathfrak C$ be the monster structure, let $P\subseteq\mathfrak C$ be the distinguished predicate defining the lovely-pair structure $\mathfrak C_P$, and let mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textnormal{}}\,\,\,\,}} be the given independence relation. Write mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textnormal{}}\,\,\,\,}}_P for its localization over $P$.

Extension conjecture for lovely pairs. There exists an independence relation mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textnormal{2}}\,\,\,\,}} on $\mathfrak C_P$ such that:

1. mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textnormal{2}}\,\,\,\,}} coincides with mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textnormal{}}\,\,\,\,}} on subsets of $P$;
2. mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textnormal{2}}\,\,\,\,}}_P=mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textnormal{}}\,\,\,\,}}_P.

The proposition immediately before the candidate shows that the localized relation mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{\scriptscriptstyle\textnormal{}}\,\,\,\,}}_P is already an independence relation on $\mathfrak C_P$, with the same local-character constant, and preserves property (*). Thus the conjectural extension is closely related to the localization construction, but the stated existence claim itself should be checked against that proposition.