5 problems

Topological nonconjugacy conjecture. The mappings $T_3$ and $T_5$ are not topologically conjugate on $\mathbb{Q}_{(2)}$.

5x+1 growth exponent conjecture. For every integer $a\not\equiv0\pmod 5$, the exponent $\eta_5(a)$ exists and equals a constant $\eta_5$ independent of $a$, with

Long-term dynamics conjecture. Every $3x+1$ trajectory eventually enters a finite cycle, whereas almost every $5x+1$ trajectory diverges.

5x+1 finite-periodic-orbit conjecture. There are finitely many periodic orbits, including

5x+1 divergent-trajectory conjecture. A density-one set of integers belongs to divergent trajectories, meaning