Rödl's maximum-degree reduction gives a linearly large induced set on which one of the two complementary graphs is $1/64^2$-sparse. In the direct orientation, iterate the sparse-house acceleration lemma until the requested parameter is reached. In the complementary orientation, the graph is $P_5$-free, so the sparse anticomplete-pair lemma gives a complete two-block blockade in the original graph. A single exponent absorbs the fixed and polynomial losses.