The normalized frame inequalities for cube sections and cross-polytope projections

Let $S=\{v_1,\ldots,v_n\}$ be an $(n,k)$-frame spanning $\mathbb{R}^k$, let

$$B_S=\left(\sum_{i=1}^n v_i\otimes v_i\right)^{-1/2},$$

and define $P=\operatorname{co}\{\pm v_1,\ldots,\pm v_n\}$ and $Q=\bigcap_{i=1}^n\{x\in\mathbb{R}^k\mid|\langle x,v_i\rangle|\leqslant1\}$. Normalized frame inequalities. The conjectured bounds are

$$C_{\Diamond}(n,k)\mathop{\rm vol}(\Diamond^k)\leqslant\frac{\mathop{\rm vol}P}{\det B_S}\leqslant\mathop{\rm vol}(\Diamond^k);$$ $$\mathop{\rm vol}(\Box^k)\leqslant\mathop{\rm vol}(Q)\det B_S\leqslant C_{\Box}(n,k)\mathop{\rm vol}(\Box^k).$$

These inequalities rewrite the preceding projection and section conjectures in frame language. Since the source does not separately establish their status beyond that of the preceding conjectures, the general assertions remain open.