Linear bound for the treewidth contraction function

Let $p,q\geqslant 1$ be integers. Define $\ell(p,q)$ to be the minimum integer such that every graph $G$ with treewidth at least $\ell(p,q)$ contains pairwise disjoint connected subgraphs $G_1,G_2,\dots,G_n$, each with $\operatorname{tw}(G_i)\geqslant p$, and contracting each of $G_1,\dots,G_n$ to a vertex produces a minor of $G$ with treewidth at least $q$. Proposed bound.

$$\ell(p,q)\leqslant O^\ast(pq).$$

The grid-minor theorem currently gives the weaker bound $\ell(p,q)\leqslant O^\ast((pq)^9)$: a $(pq)\times(pq)$ grid can be partitioned into $q^2$ copies of the $p\times p$ grid, whose contractions yield a $q\times q$ grid. Improving this polynomial bound is posed as an open problem.