Dependence of the coarse wiring profile on the choice of parameter

There are two subgraphs $X,Y$ of the $2$-dimensional integer grid such that for any $k<l$ there is an infinite subset $I\subset \mathbb{N}$ such that the $k$-wiring profile of $X$ into $Y$ grows quadratically on $I$ while the $l$-wiring profile of $X$ into $Y$ grows linearly on $I$. This resolves Question 7.8 of Barrett-Hume.