The internal development proves the inequality for a fixed linear order. The submitted parameters are minima over vertex permutations, so it is enough to run the internal bound under one well-chosen order: take a permutation witnessing $scol G r$ and order the vertices by their positions under it. Every vertex strongly reachable in the internal sense is strongly reachable in the submitted sense — a path is a walk — so the internal $scol$ under that order is at most $scol G r$; and every vertex weakly reachable in the submitted sense is weakly reachable in the internal sense — a walk bypasses to a path with a smaller support — so the submitted $wcol$, being the minimum over permutations, is at most the internal one. The right-hand side is monotone in the strong coloring number, which closes the chain.

The statement is Lemma 2.6 of Chapter 2 of the sparsity lecture notes of Pilipczuk and Siebertz (numbering of the 2019/20 edition). The internal per-order version is $Lax12Proofs.OrderedParameterBounds.wcol_le_of_scol$.