Ayad–Kihel conjecture on determining prime splitting from fixed-divisor data

Let $\mathbb{K}$ be a Galois extension of $\mathbb{Q}$ of degree $n$, and let $p$ be a prime such that $p\mid\mathfrak{J}(\mathbb{K}\mid\mathbb{Q})$. Let $\rho(p)$ denote the number of $\overline{a}\in\mathcal{O}_{\mathbb{K}}/p\mathcal{O}_{\mathbb{K}}$ such that $p\mid d(\mathbb{Z},f_a)$. Ayad–Kihel conjecture. The quantity $\rho(p)$ determines the splitting of $p$ in $\mathbb{K}$. This conjecture connects fixed divisors and the splitting behavior of primes in Galois number fields; the source gives no evidence of a resolution.