4 problems

Periodic approximation conjecture. The almost-sure limit $F(c)$ can be obtained from the periodic approximants by taking

Periodic-sequence convergence conjecture. For every $c\in[0,1]$, the sequence $\{F_X(k,c)\}$ converges, and

2-periodic convergence conjecture. For every $0\leq c\leq1$, the sequence $F_X(k,c)$ converges as $k\to\infty$; the convergence need not be monotonic unless $c=1/2$.

Monotonicity conjecture. For every $c$, the sequence $\{F_X(k,c)\}_{k\geq1}$ is monotonic.