Spread versus width conjecture for grid tree decompositions

Let $m\geq n\geq 1$ be integers, and let $a$ denote the additional width parameter in a tree-decomposition of the $(n\times m)$-grid whose width is $n+a$. The spread versus width conjecture. There is a constant $\delta>0$ such that every such tree-decomposition has a vertex with spread at least

$$\delta (n/a).$$

This would strengthen Wood's conjecture that every tree-decomposition of the $(n\times n)$-grid with width $n$ has a vertex with spread $\Omega(n)$, and would be best possible up to the multiplicative constant. The conjecture concerns the trade-off between the width and spread of tree decompositions; its status is not resolved in the supplied text.