The Ω-divergence conjecture for non-periodic E-sequences

Let $(a_n)_{n\geqslant 1}$ be an E-sequence, meaning an infinite sequence of positive integers, and define

$$b_n=\sum_{i=1}^n a_i.$$

An E-sequence is Ω-convergent to an odd positive integer $x$ if it is the E-sequence of $x$ under the $3x+1$ iteration, and it is Ω-divergent if it is not the E-sequence of any odd positive integer. The Ω-divergence conjecture. (i) Every non-periodic E-sequence is Ω-divergent; (ii) every E-sequence $(a_n)_{n\geqslant 1}$ satisfying

$$3^n>2^{b_n}\qquad\text{for all }n\in\mathbb{N}$$

is Ω-divergent. The first assertion connects periodicity of trajectories with the behavior of generalized E-sequences, while the second gives a sufficient growth condition for Ω-divergence. The source provides no resolution of either assertion.