A lower bound on the tree-width of graphs with irrelevant vertices

For their famous algorithm for the disjoint paths problem, Robertson and Seymour proved that there is a function $f$ such that if the tree-width of a graph $G$ with $k$ pairs of terminals is at least $f(k)$, then $G$ contains a solution-irrelevant vertex (Graph Minors. XXII., JCTB 2012). We give a single-exponential lower bound on $f$. This bound even holds for planar graphs.