The characterization conjecture for equal consecutive Stern degrees

Let $e(n)$ denote the degree sequence associated with the Stern polynomials, and let $\mathcal{E}$ be the set of indices at which consecutive values are equal. Let $p_n$, $q_n$, $U_i$, and $V_i$ be the recursively defined sequences and sets introduced in the paper, and define

$$\mathcal{E}'=\{2p_n\}_{n=1}^{\infty}\cup\{q_n\}_{n=1}^{\infty}\cup\bigcup_{i=1}^{\infty}(U_i\cup V_i).$$

The characterization conjecture.

$$\mathcal{E}=\mathcal{E}'.$$

The claim would give a complete description of the indices with equal consecutive degree values. It is based in part on computation through $10^8$, and no proof or resolution is supplied in the excerpt.