elementary number theory - Finitely many Supreme Primes?

A challenge on codegolf.stackexchange is to find the highest "supreme" prime: https://codegolf.stackexchange.com/questions/35441/find-the-largest-prime-whose-length-sum-and-product-is-prime

A supreme prime has the following properties:

• the number itself is prime


• the number of digits is prime


• the sum of digits is prime



• the product of digits is prime

Are there finitely many "supreme" primes? Are there infinitely many? Currently the highest one found is ~$10^{72227}$