Power-law growth conjecture for Pratt-tree avoiding prime sets

For each prime $r$, let $\mathcal{P}_r$ be the corresponding set of primes avoiding $r$ in their Pratt trees, and let

$$P(x)=\#\{p\in\mathcal{P}_r:p\leqslant x\}.$$

Growth conjecture for $\mathcal{P}_r$. For each $r$, there is a number $\delta_r>0$ such that

$$P(x)=x^{1-\delta_r+o(1)}\quad\text{as }x\to\infty.$$

This is an educated guess based partly on computations for $\mathcal{P}_3$; the paper gives no heuristic establishing the asserted exponent, and the conjecture remains open.