The Gram locus conjecture for real binary tensors

Let $n\geq 4$, and let $\mathcal{G}(\mathcal{B})$ denote the Gram locus of real binary tensors, with coordinates $d_1,\ldots,d_n$. Define

$$Q_1=\prod_{i=1}^n\left(\sum_{j\ne i}d_j-d_i\right)$$

and let $Q_2$ be the polynomial obtained from

$$Q_2=\frac{1}{2}\prod_{\{i,j,\ldots,k\}\in\{\pm1\}}\left(i\sqrt{d_1}+j\sqrt{d_2}+\cdots+k\sqrt{d_n}\right).$$

Gram locus conjecture. The Gram locus $\mathcal{G}(\mathcal{B})$ is given by

$$Q_1\geq Q_2\qquad\text{and}\qquad 0\leq d_i\leq\frac{1}{4}\quad\text{for }i=1,\ldots,n.$$

This proposes a complete semialgebraic description of the Gram locus for $n\geq4$, extending the preceding exact description in the case $n=3$. The parser supplies no evidence that the conjecture has been resolved.