Local uniformization conjecture for codimension-one foliations

Let $k$ be a field of characteristic zero and let $K/k$ be a finitely generated field extension. Let $\mathcal{F}$ be a rational codimension-one foliation of $K/k$. Given a valuation $\nu$ of $K/k$, there is a projective model $M$ of $K/k$ such that $\mathcal{F}$ is log-final at the center of $\nu$ in $M$.

Local uniformization conjecture. For every valuation $\nu$ of $K/k$, there exists a projective model $M$ of $K/k$ on which the foliation $\mathcal{F}$ is log-final at the center of $\nu$.

The theorem proved in the paper establishes this statement for $k$-rational archimedean valuations. The conjecture asks for the corresponding result for arbitrary valuations and is presented as the next step toward local uniformization of codimension-one foliations.