Ruzsa's question for minimum $U^{k-2}$-uniform $k$-AP density

For each integer $k\geq4$ and $0<\alpha<1$, let $\rho_k(\alpha)$ be the largest real number such that sufficiently $U^{k-2}$-uniform subsets of $\mathbb{Z}/N\mathbb{Z}$ of density approximately $\alpha$ have $k$-AP density at least $\rho_k(\alpha)$. Ruzsa's extended question. Let $k\geq4$. For every $C>0$,

$$\rho_k(\alpha)<\alpha^C$$

for all sufficiently small $\alpha$. The paper proves this for odd $k\geq5$ in a weaker quantitative form, while the general claim remains open for even $k$.