Conjecture for consecutive Piatetski–Shapiro primes

Fix $c_1,c_2\in(1,2)$. Let ${\mathcal{N}}^{(c)}=(\fl{n^c})_{n=1}^{\infty}$, let $p^\sharp$ denote the next larger prime after a prime $p$, and define

$$\pi(x;{\mathcal{N}}^{(c_1)},{\mathcal{N}}^{(c_2)})=\left|\left\{p\leq x:p\in{\mathcal{N}}^{(c_1)}\text{ and }p^\sharp\in{\mathcal{N}}^{(c_2)}\right\}\right|.$$

Consecutive-prime conjecture. For every fixed $\varepsilon>0$,

$$\pi(x;{\mathcal{N}}^{(c_1)},{\mathcal{N}}^{(c_2)})=\frac{x^{1/c_1+1/c_2-1}}{c_1c_2\log x}+O\left(\frac{x^{1/c_1+1/c_2-1}}{(\log x)^{3/2-\varepsilon}}\right),$$

where the implied constant depends only on $c_1,c_2$ and $\varepsilon$. The paper presents this as a heuristic conjecture based on the Lemke Oliver–Soundararajan model and a strong Hardy–Littlewood hypothesis; no proof or resolution is given.