Uniform a priori bounds conjecture for quadratic renormalizations

For $c\in\partial\mathcal{M}$ and $\eta>0$, let $p_c(z)=z^2+c$, and let $\mathcal{R}_\eta p_c$ be the rescaled first-return renormalization defined by

$$\big[\mathcal{R}_\eta p_c\big](z)=\frac{1}{\eta}\Big(p_c^{\mathfrak{q}}(\eta z+c)-c\Big),$$

where $\mathfrak{q}\equiv\mathfrak{q}(\eta z+c)$ is the first return time to the $\eta$-neighborhood of $c$. Uniform a priori bounds conjecture. With appropriate understanding, the family of renormalizations in the displayed definition is precompact uniformly over all $\eta>0$ and $c\in\partial\mathcal{M}$. Such precompactness would provide the uniform a priori bounds needed to obtain dynamical consequences from iterated renormalization; the source does not state a resolution.