Infinitude of Gauss Circle Primes

Prove that there are infinitely many integers r ≥ 1 such that C(r), the number of integer lattice points inside the circle of radius r centered at the origin, is a prime number.

Background

The function C(r) counts lattice points in a circle of radius r and is shown to be congruent to 1 modulo 4. The authors empirically paper the primes among these counts and provide heuristic arguments suggesting that the number of Gauss Circle Primes up to n grows similarly to the prime counting function, motivating a conjecture of their infinitude.