Cochran–Orr–Teichner filtration rank and nontriviality conjecture

Let $\mathcal F_n$ and $\mathcal F_{n.5}$ denote the successive terms of the Cochran–Orr–Teichner filtration of the smooth knot concordance group, indexed by $n\in\mathbb N_0$. Cochran–Orr–Teichner filtration conjecture. For any $n\in\mathbb N_0$, the quotient groups

$$\mathcal F_n/\mathcal F_{n.5}$$

have infinite rank. Moreover, for $n>0$, the groups

$$\mathcal F_{n.5}/\mathcal F_{n+1}$$

are nontrivial. These assertions describe the main remaining questions about the structure of the filtrations; the stated status is open.