Before revision:  In a subsequent and very recent work, Gartland and Lokshtanov arXiv:2007.11402 extended their ideas to C_{> t-free graphs: graphs that do not contain a cycle on more than t vertices as an induced subgraph.
After revision:  In a subsequent and very recent work, Gartland and Lokshtanov arXiv:2007.11402 -free graphs: graphs that do not contain a cycle on more than t vertices as an induced subgraph.