Finite burden for fields with finitely many valuation rings

Let $K$ be a dp-minimal pure field, and let $\mathcal{O}_1,\ldots,\mathcal{O}_n$ be valuation rings on $K$. The structure $(K,\mathcal{O}_1,\ldots,\mathcal{O}_n)$ expands $K$ by these valuation rings.

Finite-burden conjecture. The structure $(K,\mathcal{O}_1,\ldots,\mathcal{O}_n)$ has finite burden.

This asks whether the finite-burden results for real closed and $p$-adically closed fields extend to arbitrary dp-minimal pure fields. The conjecture seems likely for dp-minimal fields of characteristic zero, while positive characteristic may cause additional problems; the statement is presented as open in the source.