The conjecture on prime numbers that are palindromic in base b

Let $b\geq 2$ be an integer. A positive integer is $b$-palindromic if its sequence of base-$b$ digits is a palindrome, and let $\mathscr{P}_b(x)$ denote the set of positive $b$-palindromic integers at most $x$. Palindromic-prime conjecture. There are infinitely many prime numbers that are $b$-palindromic. In fact, for sufficiently large $x$,

$$\#\left\{p\in\mathscr{P}_b(x)\right\}\asymp_b\dfrac{\#\mathscr{P}_b(x)}{\log x}.$$

This is presented as a difficult unsolved problem concerning the expected distribution of primes among base-$b$ palindromic integers.