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# [APL (Dyalog Unicode)], 15 bytes<sup>[SBCS](https://github.com/mlochbaum/BQN/blob/master/commentary/sbcs.bqn)</sup>

With many thanks to Marshall Lochbaum and dzaima for their help debugging and golfing this answer

<!-- language-all: lang-apl -->

    {⍵∪⌊⍵*2|⍵+.5}⍣≡

```
{⍵∪⌊⍵*2|⍵+.5}⍣≡  ⍝ ⍵ is our input of either n or our intermediate results
{            }⍣≡  ⍝ run until we reach a fixed point (in this case, 1)
         ⍵+.5     ⍝ ⍵ plus one half
       2|         ⍝ mod 2 gives us 0.5 if our ⍵ was even, else 1.5
    ⍵*           ⍝ ⍵ to the power of the above exponent
   ⌊              ⍝ floor
 ⍵∪              ⍝ union of the answer with the previous results
                  ⍝ The set union will either remove an extra one, 
                  ⍝ or if we find a cycle that disproves the conjecture,
                  ⍝ will remove the extra members of the cycle
```

[Try it online!][TIO-kouflbyf]

[APL (Dyalog Unicode)]: https://www.dyalog.com/
[TIO-kouflbyf]: https://tio.run/##SyzI0U2pTMzJT///P@1R24TqR71bH3WsetTTBWRoGdUASW0909pHvYsfdS78/6hvKlBNmoKh5X8A "APL (Dyalog Unicode) – Try It Online"