Nonzero fixed-point conjecture for the hierarchical percolation renormalization map

Let $cP(cell^2_\downarrow)$ be the space of probability measures on $cell^2_\downarrow$, and let $\mathscr R=\mathscr R_{d,L,\alpha}$ be the renormalization map obtained by independently taking $L^d$ copies, applying the multiplicative coalescent for time $L^{-d-\alpha}$, and rescaling by $L^{-\frac{d+\alpha}{2}}$. Let $\mu_{\beta_c}$ be the Dirac measure supported on $(\sqrt{\beta_c},0,0,\ldots)$. Nonzero-fixed-point conjecture. If $d<3\alpha$, then

$$\mathscr R^n[\mu_{\beta_c}]$$

converges to a non-zero fixed point of $\mathscr R$. The critical orbit is known to be compact in this regime, but convergence to a nonzero fixed point is not established by the cited results and remains open.