@Sahaj I found a solution for $n=6$ (!!) : $$[31699366920, 43719523705, 107614702584]$$

Wow! That’s an amazing find.

good job!!

To clarify : I had the idea to enumerate over the gcd's which is much more efficient than enumerating over a,b,c. The gcd's uniquely determine a,b,c.

Right. So the conjecture is generally untrue I’m supposing. The cases $n=2$ and $n=3$ still seem to be of interest.

In fact, for those I did not find any solution (besides of course the case with equal entries , for which $n=a=b=c=2$ can be proven to be the only one)

I was surprised that this system even has a solution other than the "trivial" one.

yes, me too

Do you have any idea how at least the case $n=2$ could be solved ?

Here is a hot question about abundant numbers.