NP-hardness conjecture for Grothendieck d-norms

Let $d,m,n\in\mathbb{N}$. The real symmetric spaces $\mathbb{S}^n(\mathbb{R})$, real rectangular spaces $\mathbb{R}^{m\times n}$, complex symmetric spaces $\mathbb{S}^n(\mathbb{C})$, and complex rectangular spaces $\mathbb{C}^{m\times n}$ be equipped with the corresponding Grothendieck $d$-norms. NP-hardness conjecture. The following norms are all NP-hard:

1. $\lVert\,\cdot\,\rVert_{\gamma,d}^{\mathbb{R}}$ and $\lVert\,\cdot\,\rVert_{\Gamma,d}^{\mathbb{R}}$ on $\mathbb{S}^n(\mathbb{R})$ when $d(d+1)/2\le n$;
2. $\lVert\,\cdot\,\rVert_{\mathsf{G},d}^{\mathbb{R}}$ on $\mathbb{R}^{m\times n}$ when $d(d+1)/2\le m+n$;
3. $\lVert\,\cdot\,\rVert_{\gamma,d}^{\mathbb{C}}$ and $\lVert\,\cdot\,\rVert_{\Gamma,d}^{\mathbb{C}}$ on $\mathbb{S}^n(\mathbb{C})$ when $d^2\le n$;
4. $\lVert\,\cdot\,\rVert_{\mathsf{G},d}^{\mathbb{C}}$ on $\mathbb{C}^{m\times n}$ when $d^2\le m+n$.

The preceding discussion establishes NP-hardness over $\mathbb{R}$ for $d=1$; the conjecture predicts NP-hardness throughout the dimension ranges not covered by the preceding stability proposition.