Shevelev's main-increment conjecture for the dual sequence

Let $\tilde{c}(n)$ be the sequence defined by

$$\tilde{c}(1)=2,\qquad \tilde{c}(n)=\tilde{c}(n-1)+\begin{cases}\gcd(n,\tilde{c}(n-1)),&\text{if }n\text{ is odd},\gcd(n-2,\tilde{c}(n-1)),&\text{if }n\text{ is even},\end{cases}\quad n\geq2.$$

An increment is called main if it occurs at one of the distinguished fundamental points described in the paper. Shevelev's main-increment conjecture. For $n\geq16$, the main and only main increments are the record differences $\tilde{c}(n)-\tilde{c}(n-1)$. The conjecture concerns the identification of the main increments with record differences in this sequence; the supplied text gives no resolution.