Shevelev's parametrized twin-prime construction conjecture

Let $m\geq4$, set $A_m=\{1,\ldots,m-3\}$, and define

$$\tilde{c}^{(m)}(1)=m,$$

with, for $n\geq2$,

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

Let $n^*=n^*(m)$ be the point of the last nontrivial increment of $\{\tilde{c}^{(m)}(n)\}$ on $A_m$, with $n^*=1$ if there is no nontrivial increment on $A_m$. Shevelev's parametrized twin-prime construction conjecture. The numbers $\tilde{c}^{(m)}(n^*)-n^*\mp1$ are twin primes. This proposed construction is intended to produce twin primes from each integer $m\geq4$; the supplied text gives no resolution.