Remainder-class independence conjecture for prime slopes

Let $f(x,y)$ be a homogeneous polynomial of degree $n$, let $F(t)=\int_0^t[\sqrt[n]{f(1,y)}]^{-2}\,dy$, and let $\mathfrak{a}$ be an ideal in $\mathbb{Q}[\xi]$, where $\xi$ is a root of $f(x,1)$. Let $C_f^{\mathfrak{a}}(\alpha,t_1,t_2,M)$ count primes $p=f(a,b)<M$ with $a+b\xi\equiv\alpha\pmod{\mathfrak{a}}$, where $\alpha$ is invertible modulo $\mathfrak{a}$, and with $F^{-1}(t_1)<b/a<F^{-1}(t_2)$. For $t_1<t_2$ and $t_3<t_4$ in an interval where $F$ is continuous, the remainder-class independence conjecture asserts

$$\lim_{M\rightarrow\infty}\frac{C_f^{\mathfrak{a}}(\alpha,t_1,t_2,M)}{C_f^{\mathfrak{a}}(\beta,t_3,t_4,M)}=\frac{F(t_2)-F(t_1)}{F(t_4)-F(t_3)},$$

independently of the remainders $\alpha$ and $\beta$. This proposes that the normalized slope distribution is unaffected by the chosen invertible residue classes.