The quadratic-logarithmic conjecture for regular subgraphs

Let $d(r,n)$ be the smallest average degree such that every $n$-vertex graph with average degree at least $d(r,n)$ contains an $r$-regular subgraph. Quadratic-logarithmic conjecture. There exists some constant $C$ such that, whenever $r\leq \frac{1}{2}\log n$, every $n$-vertex graph with average degree at least

$$Cr^2 \log\left(\frac{\log n}{r}\right)$$

contains an $r$-regular subgraph. The paper establishes matching-order bounds for $r\geq \frac{1}{2}\log n$ and gives lower and upper bounds differing in the logarithmic factor in the stated range, so this refinement remains open.