Let $p\colon \mathbb{Z} \rightarrow \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d≥2$ with $d ∣ p(n)$ for all $n≥1$. Is it true that, for all sufficiently large $m$, there exist integers $1≤n_1<\dots < n_k$ such that $$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$$ and $$m=p(n_1)+\cdots+p(n_k)$$?