Conjecture on bounded-size regular subgraphs

Let $s\geq 3$ be an integer and let $\varepsilon>0$. Bounded-size regular-subgraph conjecture. There is a positive integer $t$ such that, for all sufficiently large $n$, every $n$-vertex graph $G$ of average degree at least $n^{1-\frac{2}{s}+\varepsilon}$ contains an $s$-regular subgraph on at most $t$ vertices. This is a regular-subgraph strengthening of the bounded-size dense-subgraph problem. The source states that it is known when $s$ is even or when $s=3$, but leaves the general case open.