Polynomial bound conjecture for excluding grasped clique minors

Let $f$ and $g$ be polynomials. For integers $r,p\geq 0$, set

$$R=f(r,p)+g(p).$$

Let $G$ be a graph and let $W$ be an $R$-wall in $G$. A flat wall is a wall that is flat in the relevant surface decomposition, and a $W$-central $a3$-decomposition is one satisfying the centrality condition defined for $W$. Polynomial bound conjecture. There exist polynomials $f$ and $g$ such that either $G$ has a model of a $K_p$ minor grasped by $W$, or there exist a set $A\subseteq V(G)$ with $|A|\leq g(p)$, a surface $\Sigma$ of Euler genus at most $g(p)$, a $W$-central $\Sigma$-decomposition $\delta$ of $G-A$ of depth and breadth at most $g(p)$, and an $r$-subwall $W'$ of $W$ such that $V(W')\cap A=\emptyset$ and $W'$ is flat in $\delta$.

The conjecture proposes replacing the exponential dependence in the theorem's current bound by polynomial functions. It concerns the quantitative structure available when a graph does not contain a clique minor grasped by a sufficiently large wall; the theorem preceding it establishes the corresponding alternative with an exponential bound.