Generalized nonintegrality conjecture for reciprocal power sums

Let $f(x)$ be a nonzero polynomial with integer coefficients and let $\mathcal{S}=\{s_i\}_{i=1}^{\infty}$ be an infinite sequence of positive integers, not necessarily increasing or distinct. For each $n$, write $\mathcal{S}_n=(s_1,\ldots,s_n)$, and for $1\leq k\leq n$ define

$$H_f^{(k)}(\mathcal{S}_n):=\sum_{1\leq i_1<\cdots<i_k\leq n}\prod_{j=1}^k\frac{1}{f(a_{i_j})^{s_{i_j}}},$$

where $Z_f=\{x\in\mathbb{Z}:f(x)=0\}$, $\{a_i\}_{i=1}^{\infty}=\mathbb{Z}^+\setminus Z_f$ is in increasing order, and

$$\bar H_f^{(k)}(\mathcal{S}_n):=\sum_{1\leq i_1\leq\cdots\leq i_k\leq n}\prod_{j=1}^k\frac{1}{f(a_{i_j})^{s_{i_j}}}.$$

Generalized nonintegrality conjecture. There is a positive integer $N$ such that for every integer $n\geq N$ and every integer $k$ with $1\leq k\leq n$, neither $H_f^{(k)}(\mathcal{S}_n)$ nor $\bar H_f^{(k)}(\mathcal{S}_n)$ is an integer.

This conjecture generalizes the two earlier conjectures cited in the source and extends the nonintegrality problem to arbitrary nonzero integer polynomials and arbitrary positive-integer exponent sequences. Its status is not established in the supplied text.