The asymptotic count of dismal primes

Let $\pi_b(k)$ denote the number of base $b$ dismal primes with $k$ digits. Prime-counting conjecture. As $k$ tends to infinity,

$$\pi_b(k)\sim (b-1)^2b^{k-2}.$$

The conjecture predicts that asymptotically almost all $k$-digit numbers satisfying the necessary digit conditions for primality are dismal primes. It is motivated by the numerical data presented in the paper, but no proof or resolution is given.