The characterization of real valuations with linear growth

Let $X$ be a projective variety and let $v$ be a real valuation on its function field, centered on $X$. The valuation $v$ has linear growth if the maximal vanishing order of sections of multiples of any big line bundle grows at most linearly with the multiple. A divisorial valuation is a valuation proportional to the order of vanishing along a prime divisor on a birational model of $X$.

Linear-growth characterization. A real valuation $v$ has linear growth if and only if there exists a divisorial valuation $v'$ such that $v\leqslant v'$ at the center of $v$ on some birational model.

This would characterize linear growth in terms of domination by a divisorial valuation. The preceding result establishes that every Abhyankar valuation has linear growth, while the paper notes that the converse fails; the status of the proposed characterization is not specified in the source.