Linnik-prime representation conjecture for three-variable Piatetski–Shapiro equations

Let $c_0>1$. For fixed $c$ with $1<c<c_0$ and every sufficiently large positive integer $N$, consider the equation

$$[p_1^c]+[p_2^c]+[p_3^c]=N,$$

where $p_1,p_2,p_3$ are prime numbers and each $x_i,y_i$ is an integer. Linnik-prime representation conjecture. The equation has a solution in prime numbers $p_1,p_2,p_3$ such that

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

The preceding theorem proves an asymptotic formula for a weighted count when $1<c<\frac{16559}{15276}$, but does not establish this stronger simultaneous representation claim for any stated interval up to $c_0$; the challenge is therefore left open.