Spanning subdivision conjecture for dense graphs

For $d700>0$, there is a positive constant $C_0$ such that, for every $C\ge C_0$ and $m\in\mathbb N$, the following holds. Let $G$ be a graph on $N=Cm$ vertices, and let $\delta(G)$ denote its minimum degree.

Spanning subdivision conjecture. If

$$\delta(G)\ge (1+\varepsilon)\frac{N}{2},$$

then $G$ contains a spanning $H$-subdivision for every graph $H$ with $m$ edges and no isolated vertices.

This is a dense-host extension of the paper's spanning-subdivision results. The supplied text states it in a conjecture environment but gives no resolution evidence.