We call a graph $H$ $D$-balanced (or $D$-almost-regular) if the maximum degree of $H$ is at most $D$ times the minimum degree of $H$.Is it true that for every $m\ge 1$, if $n$ is sufficiently large, any graph on $n$ vertices with $\ge n\log n$ edges contains a $O(1)$-balanced subgraph with $m$ vertices and $\gg m\log m$ edges (where the implied constants are absolute)?