The colour of numbers II

Consider all natural numbers, i.e. \(\{0,1,2,\dots\}\), and suppose you are to colour each of these numbers using no more than \(k\) different colours (\(k\in\mathbb{N}\)).
Is it possible to colour all numbers, such that for each two different numbers, their sum and product have different colours?

Source: inspired by Prime's PUMA 2013.

As stated in the previous problem, this is not possible when using only three colours. Using similar arguments, one can also prove this is not possible when using four, five... colours.

However, the solution to this problem remains unknown.

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