A constructive proof shows there are at most (2r+1)^{2^{k-1}-1} progressing Splitter moves if the radius-r game can be won in k rounds on a nowhere dense graph.
Canonical decompositions in monadically stable and bounded shrubdepth graph classes
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A Note on Constructive Canonical Splitter Strategies in Nowhere Dense Graph Classes
A constructive proof shows there are at most (2r+1)^{2^{k-1}-1} progressing Splitter moves if the radius-r game can be won in k rounds on a nowhere dense graph.