Inverse-eigenvector product conjecture for Gram matrices

Let $M$ be a real $n\times n$ Gram matrix with diagonal entries equal to $1$. A vector $\alpha=(\alpha_i)_{i=1}^n$ is an inverse eigenvector of $M$ if

$$M\alpha=\alpha^{-1},\qquad \alpha^{-1}=\left(\frac1{\alpha_1},\dots,\frac1{\alpha_n}\right).$$

Inverse-eigenvector product conjecture. There exists an inverse eigenvector $\alpha$ such that

$$\left|\prod_{i=1}^n\alpha_i\right|\leqslant1.$$

The source states that this linear-algebraic formulation implies the original real polarization problem. The existence assertion is left as a conjecture.