The prime-power annihilation conjecture for polynomial rings over the integers

Let $R$ be a polynomial ring over the integers, and let $f_i,g_i$ be elements of $R$ satisfying

$$f_1g_1+\dots+f_ng_n=0.$$

Prime-power annihilation conjecture. For every prime power $q=p^e$, there exists $k\in\mathbb{N}$ such that

$$\frac{(f_1g_1)^q+\dots+(f_ng_n)^q}{p}(g_1\cdots g_n)^k\in(g_1^{q+k},\dots,g_n^{q+k})R.$$

This conjecture is motivated by the problem of controlling torsion in local cohomology over polynomial rings over the integers and is presented as an open consequence-related problem in the source.