Polynomial pivot-minor Ramsey number conjecture

Let $R_{\mathrm{piv}}(k)$ be the minimum $n$ such that every $n$-vertex graph contains an independent set or clique of size $k$ as a pivot-minor. Pivot-minor Ramsey conjecture.

$$R_{\mathrm{piv}}(k) \leq \mathsf{poly}(k).$$

This conjecture would imply the polynomial bound for the vertex-minor Ramsey number. It holds when the graphs are restricted to any proper pivot-minor closed class, but remains open for arbitrary graphs.