Vatwani's Gaussian prime pair conjecture

Let $m_1,m_2$ be integers of the same parity. Vatwani's Gaussian prime pair conjecture. There are infinitely many pairs of rational primes $(p_1,p_2)$ of the form

$$p_1=a^2+b^2,\qquad p_2=(a+m_1)^2+(b+m_2)^2.$$

The conjecture is a Gaussian-integer analogue of bounded-gap questions for rational primes. The source attributes it to Akshaa Vatwani in 2017 and gives no evidence of a resolution.