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Extreme disproportion in the dispersion of "Digital Roots" of the highest numbers reach in the Collatz Conjecture.

I calculated the Digital Root remainder mod 9 for the highest numbers reached for each seed in Collatz Conjecture: for example, for $39$ as the seed number, the highest number reached is $304$, so the remainder mod 9 is $7$.
What I found is that there is a huge disporportion in the numbers obtained, most of them being $7$: in the first $100$ seed numbers, $65$ of them give $7$ as remainder mod 9.
I checked the pattern even for random numbers: they mostly led to $7$ (for example $111$, $222$, $333$, $444$, $555$, $666$, $777$, $888$, $999$ all lead to $7$).
For a normal set of natural numbers the remainder mod 9 would be proportionally distributed between $1$ and $9$, so I cannot find any reason for this strange behavior.

My questions.
Can anyone explain this phenomenon? Or is there any paper/monograph about it? Has anyone seen this phenomenon before?

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    $\begingroup$ en.wikipedia.org/wiki/Digital_root The digital root $\text{dr}_{b}(n)$ of a positive integer $n$ satisfies $\text{dr}_{b}(n)\equiv n\mod b-1$, so this question would be be better received if it were phrased in terms of modular arithmetic. This phenomenon also occurs modulo $3^{n}$ which is not too surprising. $\endgroup$
    – Joseph Van Name
    Commented Nov 20, 2021 at 12:13

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Well, digital root function is just the number modulo $9$ and Collatz map naturally doesn't produce anything uniform modulo $9$. To answer your question, look at the maximal point in the orbit of a number. Naturally it is a number of the form $3x+1$ obtained from some $x$. Now, $x$ itself is of the form $(3y+1)/2^t$ for some $t\geq 1$. If $t\geq 2$ then we have $y>3x+1$ so $3x+1$ is not a maximum. Hence, $x=(3y+1)/2$ and so the maximal number has the form $3((3y+1)/2)+1=9y/2+5/2$ and so it is, indeed, $7$ modulo $9$.

I cheated a bit since it may happen that the maximal number is the number you start from, or the second one, or the third one, and then the reasoning above does not apply. But one may expect that for most numbers the maximum in the orbit is not among first three numbers.

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