de Klerk–Laurent hypercube quadratic-module degree conjecture

Let $n\in\mathbb{N}$ be even, and let $\mathrm{B}^{n}=[-1,1]^n$. Write $\mathcal{Q}(\mathrm{B}^{n})_n$ for the degree-$n$ truncation of the quadratic module generated by $1-x_1^2,\ldots,1-x_n^2$. de Klerk–Laurent's conjecture.

$$(1-x_1^2)(1-x_2^2)\ldots(1-x_n^2)+\frac{1}{n(n+2)}\in\mathcal{Q}(\mathrm{B}^{n})_n.$$

This conjecture would give a substantially sharper effective form of Putinar's Positivstellensatz on the hypercube than the general bounds currently available. It was proposed by de Klerk and Laurent and remains open.