The Standard Model effective field theory (SMEFT) is a robust model for studying beyond Standard Model physics. The dimension-eight operators are of particular interest to theoretically rule-out regions of Wilson coefficient (WC) space based on positivity bounds derived using properties of scattering amplitudes. The renormalisation group evolution (RGE) of these WCs is necessary to define them consistently at different scales. In this work, we present the renormalisation of the bosonic sector of the SMEFT from tree-level generated dimension-eight operators up to $O (v^4/\Lambda^4)$, with v $\sim$ 246 GeV as the electroweak symmetry breaking scale and $\Lambda$ as the cut-off scale of the EFT. Using these RGEs, and the positivity bounds on $ X^2 \phi^2 D^2 $ class, we deduce the constraints on WCs of $ \phi^4 D^4$ and $ \psi^2 \phi^2 D^3 $ classes. These bounds on $ X^2 \phi^2 D^2 $ operators remain valid at sufficiently small scales within the one-loop analysis.