Singh's polynomial-ring conjecture on p-torsion in local cohomology

Let $R$ be a polynomial ring over the integers, and let $F_i,G_i\in R$ satisfy

$$F_1G_1+\dots+F_nG_n=0.$$

For a prime integer $p$ and a prime power $q=p^e$, define

$$\lambda_q=\frac{(F_1G_1)^q+\dots+(F_nG_n)^q}{p}.$$

Singh's conjecture. There exists $k\in\mathbb N$ such that

$$\lambda_q(G_1\cdots G_n)^k\in(G_1^{q+k},\dots,G_n^{q+k})R.$$

Equivalently, the associated candidate $p$-torsion class vanishes after a sufficiently high transition in the direct-limit description of local cohomology. The conjecture is established when $n=2$, but the supplied text does not state a resolution in the polynomial-ring case for general $n$.