B-regular refinement of divisor-counting estimates

Let $\mathcal P(g,t)$ and $\mathcal Q_{\alpha,\beta}$ be the prime sets defined earlier in the paper, and let $\delta(g,t)$ and $\rho_{\alpha,\beta}(a,d)$ be the corresponding constants in the stated estimates. B-regular refinement conjecture. If the primes counted by $\mathcal P(g,t)$ and $\mathcal Q_{\alpha,\beta}$ are additionally required to be $B$-regular, then the estimates remain valid after replacing

$$\delta(g,t)\text{ by }\frac{\delta(g,t)}{\sqrt e},\qquad \rho_{\alpha,\beta}(a,d)\text{ by }\frac{\rho_{\alpha,\beta}(a,d)}{\sqrt e}.$$

The conjecture applies the Siegel heuristic for $B$-regular primes to these divisor-counting estimates; the resulting simultaneous distribution statement is not proved.