Generic complete-graph rigidity conjecture for admissible matrix norms

Let $X$ be either ${\mathcal{M}}_n({\mathbb{F}})$ or ${\mathcal{H}}_n({\mathbb{F}})$, let $\|\cdot\|$ be an admissible norm on $X$, and write $k=\dim X$. For a placement $p\in X^V$, the framework $(K_m,p)$ is called full, well-positioned, and infinitesimally rigid when it has the corresponding properties in $(X,\|\cdot\|)$. Complete-graph rigidity conjecture. If ${\mathbb{F}}={\mathbb{R}}$, then there exists $p\in X^V$ such that $(K_m,p)$ is full, well-positioned, and infinitesimally rigid for all $m\geq 2k$; if ${\mathbb{F}}={\mathbb{C}}$, then there exists such a placement for all $m\geq 2k-1$. The conjecture predicts the first values of $m$ beyond the Maxwell-counting obstructions for complete graphs in admissible matrix spaces; its status is not resolved in the supplied source context.