Rationality conjecture for limiting invariants of Fermat hypersurface rings

For $d\geq2$, let

$$S_{p,n,d}=\mathbb{F}_p[[x_0,\ldots,x_n]]/(x_0^d+\cdots+x_n^d)$$

be the degree-$d$ Fermat hypersurface ring, and define $c_n$ and $c'_n$ by

$$\lim_{p\to\infty}e_{HK}(S_{p,n,d})=c_n,$$ $$\lim_{p\to\infty}s(S_{p,n,d})=c'_n.$$

Here $\xi_{2d}$ is a primitive $2d$-th root of unity. Rationality conjecture. The generating functions

$$\sum_{n\geq0}c_n\alpha^n\qquad\text{and}\qquad\sum_{n\geq0}c'_n\alpha^n$$

are rational functions in $\alpha$, $\cos(\lambda_i\alpha)$, and $\sin(\lambda_i\alpha)$, with the coefficients of the rational functions and the $\lambda_i$ lying in $\mathbb{Q}(\xi_{2d})$. This conjecture seeks a uniform description of the limiting Hilbert–Kunz multiplicities and $F$-signatures for Fermat hypersurface rings across dimensions. The supplied source gives no resolution status.