The lower-bound conjecture for secure domination in subdivisions

Let $G$ be a graph, and let $G^{\frac{1}{2}}$ denote its $2$-subdivision, obtained by replacing each edge with a path of length $2$. Write $\gamma_s(H)$ for the secure domination number of a graph $H$.

Secure-domination lower-bound conjecture. For every graph $G$,

$$\gamma_s\left(G^{\frac{1}{2}}\right)>\frac{4}{5}|V(G)|.$$

The conjecture is motivated by examples showing that the secure domination number of a $2$-subdivision can be smaller than both the number of edges and the number of vertices of the original graph; its status is not established in the supplied source.