  We calculate the renormalisation group running of the bosonic Standard Model
(SM) effective operators at one loop and to order $v^4/\Lambda^4$, with $v\sim
246$ GeV being the electroweak scale and $\Lambda$ the unknown new physics
threshold. We focus on contributions driven by one dimension-eight term and SM
couplings, thus extending (and completing) the effort initiated in
arXiv:2106.05291, in which quantum corrections from pairs of dimension-six
interactions were considered. We highlight some interesting consequences,
including the renormalisation of loop-induced interactions by tree-level
generated terms and, more importantly, the validity of positivity bounds on
different operators inducing anomalous gauge quartic couplings.
