Five-prime Linnik representation conjecture for Diophantine inequalities

Let $\varepsilon>0$ be a small constant. There exists $c_0>1$ such that for any fixed $1<c<c_0$ and every sufficiently large positive number $N$, one seeks primes $p_1,p_2,p_3,p_4,p_5$ and integers $x_i,y_i$ satisfying

$$\left|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N\right|<\varepsilon$$

with

$$p_i=x_i^2+y_i^2+1\qquad (1\leq i\leq 5).$$

Five-prime Linnik representation conjecture. For every fixed $1<c<c_0$ and every sufficiently large positive number $N$, the displayed Diophantine inequality has a solution in prime numbers $p_1,p_2,p_3,p_4,p_5$, with each prime of the form $p_i=x_i^2+y_i^2+1$.

This is posed as a future task after the paper proves a weaker result in which only $p_1$ is required to have the form $x^2+y^2+1$, with an explicit bound on $c$ and a shrinking error term. The simultaneous requirement that all five primes be Linnik primes is therefore left open in the source.