[seqfan] primes beginning 101 in binary

[Neil Sloane]

Alois has produced a clean binary version of A089755 and A262283, namely
A262350:
a(1) = 2. For n>1, let s denote the binary string of a(n-1) with the
leftmost 1 and following consecutive 0's removed. Then a(n) is the smallest
prime not yet present whose binary representation begins with s.

This led me to look at "primes beginning 101 in binary" etc.:
Primes whose binary expansion begins with binary expansion of 1, 2, 3, 4,
5, 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.

My question is, is it known that any of these sequences are infinite (other
than A000040, the primes themselves)?

Is there a proof that the new sequence A262350 or its complement are
infinite?