4 problems

Oscillation conjecture. There exists a constant $C>0$ such that, for every positive integer $k$, there is a pair of binary words $(v,w)$ with lengths at most $\exp(Ck)$ for which

Positive-density conjecture. The limit

Optimal-pair conjecture. For $k\in\{1,2,\ldots\}$ and $n$ sufficiently large, there exists $r=r(n,k)$ such that the maximizing pairs in $W(n,k)$ are exactly the pairs specified by…

Monotonicity conjecture. The number of pairs $(v,w)$ satisfying