Brinkmann, Preissmann and Sasaki's large-girth Type 2 conjecture

Let $G$ be a cubic graph, meaning that every vertex has degree $3$, and let its girth be the length of its shortest cycle. A cubic graph is Type 2 when its total chromatic number is $5$, namely, $5$ colors are required, equal to its maximum degree plus two. Brinkmann, Preissmann and Sasaki's conjecture. There is no Type 2 cubic graph with girth at least $5$.

The conjecture arose from the unsuccessful search for Type 2 snarks of girth at least $5$; it separates the large-girth and snark requirements. No resolution is reported in the source.