Finite burden after adding a valuation ring to a field of finite burden

Let $K$ be a field of finite burden, possibly with extra structure, and let $\mathcal{O}$ be a valuation ring on $K$. The structure $(K,\mathcal{O})$ expands $K$ by this valuation ring.

Valuation-ring finite-burden conjecture. The structure $(K,\mathcal{O})$ has finite burden.

This is a broad generalization of the preceding conjectures, allowing extra structure on $K$ and only one valuation ring. The source gives no resolution and presents it as a general conjectural extension of the finite-burden phenomenon.