Extension to Sharky Kesa's problem

If you haven't seen this problem yet, do try it out. Spoilers follow.

I would like to propose an extension: is it possible to improve the bound $O(n!^3)$ ?

An idea I had is start with a primorial cubed, multiply by 8 each step, until the next primorial cubed is needed. Of course, this would allow arbitrarily large subsequences of geometric sequences, so we will not allow increments of 8, ie $\dfrac{t_{n+1}}{t_n}$ is strictly monotonically increasing.

#NumberTheory

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$