Siegel's asymptotic conjecture for B-regular primes

Let $R_B(x)$ denote the number of $B$-regular primes up to $x$, where an odd prime is $B$-regular if it divides none of the numerators of $B_2,B_4,\ldots,B_{p-3}$. Siegel's conjecture. As $x$ tends to infinity,

$$R_B(x)\sim \frac{x}{\sqrt{e}\log x}.$$

This heuristic predicts the observed proportion of regular primes, approximately $e^{-1/2}$, but the asymptotic formula is not known.