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on The complex Grothendieck constant ($C_{10b}$)

The statement

The complex Grothendieck constant (often denoted $K_G^{\mathbb{C}}$ ) is the smallest number $C_{10b}$ such that, for every $m,n\ge 1$ and every complex matrix $A=(a_{ij})\in\mathbb{C}^{m\times n}$ ,

\max_{\substack{u_1,\dots,u_m\in S^{\infty}\\ v_1,\dots,v_n\in S^{\infty}}} \left|\sum_{i=1}^m\sum_{j=1}^n a_{ij}\langle u_i, v_j\rangle\right| \ \le\ C_{10b}\ \max_{\substack{|s_1|=\cdots=|s_m|=1\\ |t_1|=\cdots=|t_n|=1}} \left|\sum_{i=1}^m\sum_{j=1}^n a_{ij}s_it_j\right|.

Here $S^{\infty}$ denotes the unit sphere in a (complex) Hilbert space, $\langle\cdot,\cdot\rangle$ is the Hermitian inner product, and $t_{1}, \ldots, t_{n}, s_{1}, \ldots, s_{m}$ are complex numbers.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.