Faudree–Gyárfás–Schelp star-forest ascending decomposition conjecture

Let $G$ be a graph with $\binom{m+1}{2}$ edges. An ascending subgraph decomposition is a decomposition $H_1,\dots,H_m$ in which $H_i$ has $i$ edges and is a subgraph of $H_{i+1}$ for each $i$. A star forest is a forest whose connected components are stars. Faudree–Gyárfás–Schelp conjecture. Every graph $G$ with $\binom{m+1}{2}$ edges has an ascending subgraph decomposition $H_1,\dots,H_m$ in which every $H_i$ is a star forest. The paper establishes the broader ascending decomposition conjecture but leaves open whether the decomposition can always be chosen entirely from star forests.