It's not 6 times higher, but 9 times

Honestly I don't dare to say, I'm sorry.
I expect it to use at least 3x as much memory, but that's a wild guess really.
It depends on so many things like render detail and GI settings.
There are 2 things you could try, if you have the time:
1) I don't know if it will represent it very well, but you may consider rendering some crops of the current resolution vs the final resolution and see how the memory consumption changes.
2) Compare memory consumption when rendering full image @ 8400x2400 vs 4200x1200 vs 2100x600 vs 1050x300 and plot the RAM usage in something like Excel. You could set up an animation to do this all in 1 night. On your new machine it will take just a couple of hours, so you may even consider to add a couple of resolution just in between.
I'll attach an Excel later...