Equality of the asymptotic clique bounds for bounded maximum average degree

For each positive integer $k$, let $\alpha_k$ be the minimum value such that there is a constant $c_k$ with

$$\omega(G^2)\leqslant \alpha_kD+c_k$$

whenever $G$ is $k$-degenerate and has maximum degree at most $D$. Let $\beta_k$ be the minimum value such that there is a constant $d_k$ with

$$\omega(G^2)\leqslant \beta_kD+d_k$$

whenever $G$ has maximum average degree less than $2k$ and maximum degree at most $D$. The equality conjecture states that, for all $k\geqslant 2$,

$$\beta_k=\alpha_k.$$

The paper proves the equality for $k=2$, while its validity for larger values of $k$ remains open; the authors do not conjecture the precise values of either parameter.