Ternary Diophantine approximation conjecture for primes of the form $x^2+y^2+1$

Let $\varepsilon>0$ be a small constant. Let $p_1,p_2,p_3$ be prime numbers and let $x_i,y_i$ be integers for $1\leq i\leq 3$. The primes are required to have the form

$$p_i=x_i^2+y_i^2+1\qquad (1\leq i\leq 3).$$

Conjecture on ternary approximation. There exists $c_0>1$ such that, for any fixed $1<c<c_0$ and every sufficiently large positive number $N$, the inequality

$$|p_1^c+p_2^c+p_3^c-N|<\varepsilon$$

has a solution with each $p_i$ prime and of the prescribed form.

The conjecture strengthens the theorem proved in the paper, which allows only $p_1$ to be of the form $x^2+y^2+1$ and gives a shrinking error for $1<c<427/400$. It asks whether all three primes can satisfy this representation while approximating every sufficiently large target arbitrarily closely for a fixed small $\varepsilon$; no resolution is supplied here.