Puzzle 1006. Bemirps as index of the π(i), the prime counting function.

G. L. Honaker, Jr. made me notice the following curio by Sariyar:

"The smallest bemirp such that 3 of the 4 associated primes are prime-indexed primes: π(1888691180161) = 69368137291 is prime, π(1888961180191) = 69377688539 is prime, π(1910811698881) = 70150532951 is prime. Unfortunately, π(161081196881) = 59510529369 = 3^2 × 883 × 7488427. [Sariyar]"

A bemirp (or bi-directional emirp) is a prime that yields a different prime when turned upside down with reversals of each being two more different primes. The result is four different associated primes. E.g., 1061 turned upside down yields 1901, and the reversals of these are 1601 and 1091 respectively.

Sariyar discovered that 1888691180161 is a bemirp because

the following four are primes:

1888691180161

1888961180191

1910811698881

1610811968881

Morover, Sariyar discovered the π(x) is prime when x is sustituted by the first three values:

π(1888691180161) = 69368137291 is prime
π(1888961180191) = 69377688539 is prime
π(1910811698881) = 70150532951 is prime.

But

π(161081196881) = 59510529369 is a composite equal to
3^2 × 883 × 7488427

Q. Can you find a bemirp whose four π counting functions are primes?