Generalized nonintegrality conjecture for multiple reciprocal sums of polynomials

Let $f(x)$ be a nonzero polynomial of integer coefficients. Let $Z_f:=\{x\in\mathbb{Z}:f(x)=0\}$, let $\{a_k\}_{k=1}^{\infty}:=\mathbb{Z}^+\setminus Z_f$ be arranged in increasing order, and define

$$M_{k,f}(\vec{s},n):=\sum_{1\leq i_1<\cdots<i_k\leq n}\prod_{j=1}^k\frac{1}{f(a_{i_j})^{s_j}},$$ $$M_{k,f}^*(\vec{s},n):=\sum_{1\leq i_1\leq\cdots\leq i_k\leq n}\prod_{j=1}^k\frac{1}{f(a_{i_j})^{s_j}}.$$

Let $\{s_i\}_{i=1}^{\infty}$ be an infinite sequence of positive integers, and write $\vec{s}^{(k)}:=(s_1,\ldots,s_k)$. 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 $M_{k,f}(\vec{s}^{(k)},n)$ nor $M_{k,f}^*(\vec{s}^{(k)},n)$ is an integer. The sequence need not be increasing or have distinct terms. This conjecture generalizes Conjecture 3.1 of the cited work and extends the expected eventual nonintegrality of multiple reciprocal sums and star sums to arbitrary nonzero integer polynomials, including polynomials with integer roots.