Here is a new conjecture of mine from the appendix of an unpublished manuscript currently under review.
Let $b \in \mathbb{Z}^+$ and assume that $n$ is an integer greater than $1$ and not a multiple of $10$.
Let $^b{n}$ be $n^{n^{{\cdots}^n}}$ ($b$-times $-$ see tetration).
Then, let us call $\bar{b}$ the smallest hyperexponent of $n$ such that the congruence speed of $n$ is constant (i.e., $\bar{b} := \bar{b}(n)$) and matches the result of Equation (16) of This Paper.
Finally, in radix-$10$, we define as asymptotic phase shift of $n$ (APH$(n)$) the $n$-th term of the OEIS sequence A376842 (i.e., A376842$(n)$), where, if $n$ is not a multiple of $10$, the value of the most significant digit of each A376842$(n)$ corresponds to the congruence class modulo $10$ of the difference between the rightmost non-stable digit of $^\bar{b}{n}$, say the $m$-th by counting digits from right to left, and the $m$-th rightmost digit of $^{\bar{b}+1}{n}$ (which is always a stable digit since the constant congruence speed of $n$ necessarily belongs to $\mathbb{Z}^+$).
E.g., if $n = 57$ is given, then we calculate APS$(57)$ by looking at the following sequence of iterations:
$\ldots 00000000000000000000000000057$ (height $b = 1$),
$\ldots 86023081377754367704855688057$ (height $b = 2$),
$\ldots 44346417520761079004879688057$ (height $b = 3$),
$\ldots 66644650461101501156879688057$ (height $b = 4$),
$\ldots 67151164512247797156879688057$ (height $b = 5$),
$\ldots 19095327043455797156879688057$ (height $b = 6$),
$\ldots 32865132027455797156879688057$ (height $b = 7$),
$\ldots 93365364027455797156879688057$ (height $b = 8$),
$\ldots$
Since the constant congruence speed of $57$ is $3$ and $\bar{b}(57)=1$, we can finally state that APS$(57) = 2684 = A376842(57)$.
Question: Does the following conjecture hold for every $n \in \mathbb{N}-\{1\} : n \not\equiv 0 \pmod{10}$?
Conjecture: For any given $n \in \mathbb{N}-\{1\} : n \not\equiv 0 \pmod{10}$, A376842$(n) \in$ $\{2, 4, 5, 6, 8, 9, 19, 28, 46, 64, 82, 1397, 1793, 2486, 2684, 3971, 4268, 4862, 6248, 6842, 7931, 8426, 8624\}.$
Note that each term of the set
$\{2, 4, 5, 6, 8, 9, 19, 28, 46, 64, 82, 1397, 1793, 2486, 2684, 3971, 4268, 4862, 6248, 6842, 7931, 8426, 8624\}$ corresponds to some value of $n$ (a direct proof is included in Section 3 of the mentioned manuscript). Moreover, if $2 \mid n$ or $5 \mid n$ (and $n \neq 5$) we have that $\bar{b}$ does not exceed $3$ while APS$(n) \not\in \{9,19,1397,1793,3971,7931,\}$.
