Bollobás--Eldridge--Catlin conjecture for bounded-degree spanning subgraphs

Let $\Delta$ be a positive integer. Let $G$ and $H$ be $n$-vertex graphs, and write $\delta(G)$ for the minimum degree of $G$ and $\Delta(H)$ for the maximum degree of $H$.

Bollobás--Eldridge--Catlin conjecture. If

$$\delta(G)\geq \frac{\Delta}{\Delta+1}n$$

and $\Delta(H)\leq \Delta$, then $H$ is a spanning subgraph of $G$.

The conjecture remains open; the source notes that a proof for sufficiently large $n$ was announced by Kun in 2009, but no manuscript had appeared.