mathematics - IX-NAY on the IX-SAY

Will this sequence ever have a 6 in it?

9, 1, 1, 1, 10, 3, 1, 1, 10, 5, 1, 1, 10, 1, 5, 2, 1, 1, 10, 1, 1, 1, 5, 4, 1, 1, 10, 3, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10, 5, 1, 1, 5, 3, 1, 1, 5, 4, 1, ...

Daily hint #1:

The title gives away a lot, but as always, we are looking for a proof, which turns out to be surprisingly elusive. In fact, I just found a hole in my own proof, too. Oops. Well, at least the answer is still the same :-)

Daily hint #2:

@phenomist has already correctly identified the sequence, and the role of the first number in it.

It turns out that if you choose a different starting number, you get a different sequence, and sometimes, a different answer altogether! For example, here's what happens if you start with a 14:

14, 1, 10, 1, 1, 1, 5, 1, 1, 1, 10, 3, 1, 1, 5, 3, 1, 1, 10, 5, 1, 1, 5, 5, 1, 1, 10, 1, 5, 2, 1, 2, 5, 2, 1, 1, 10, 1, 1, 1, 5, 5, 1, 1, 5, 4, 1, 1, 10, 3, 1, 2, 5, 2, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10, 6, ...

Daily hint #3:

I was planning to put my code for generating these sequences on tio.run for today's hint, but alas, I've used a CPAN module that's not available there. In case you have perl5 installed, and have the means to install the missing module (Math::Roman), here's the code on pastebin.

Also, there's another hint embedded in the code itself: it outputs a newline after every occurrence of a "10". That might prove helpful :-)

Final hint: (edited to make it even hintier; the bounty is running out of time)

Here's the start of the sequence with a lot of extra formatting:

 *: 9, 
 A: 1, 1, 1, 10,
 B: 3, 1, 1, 10,
 C: 5, 1, 1, 10,
 D: 1, 5, 2, 1, 1, 10,
 A: 1, 1, 1, 5, 4, 1, 1, 10,
 B: 3, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10,
 C: 5, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 10,
 D: 1, 5, 2, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10,
 A: 1, 1, 1, 5, 4, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 10,

 B: 3, 1, 1, 5, 1, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10,
 C: 5, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 10,
 D: 1, 5, 2, 1, 1, 5, 5, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 4, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10,
 A: 1, 1, 1, 5, 4, 1, 2, 5, 3, 1, 1, 5, 4, 1, 1, 5, 5, 1, 2, 5, 2, 1, 1, 5, 1, 1, 1, 5, 2, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 10,
 B: 3, 1, 1, 5, 1, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 2, 5, 3, 1, 1, 5, 4, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 4, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10,
 C: 5, 1, 1, 5, 3, 1, 1, 5, 5, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 5, 5, 1, 2, 5, 2, 1, 1, 5, 1, 1, 1, 5, 2, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 10,
 D: 1, 5, 2, 1, 1, 5, 5, 1, 2, 5, 3, 1, 1, 5, 4, 1, 1, 5, 5, 1, 2, 5, 2, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 4, 1, 2, 5, 3, 1, 1, 5, 4, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 4, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10