Let $W$ be an infinite word over a finite alphabet $\{1,\dots,n\}$ and $k$ a positive integer. An easy application of Van der Waerden's theorem implies the existence of $k$ disjoint and consecutive intervals in $W$ such that the sum of letters in each interval are equal (**Edit**. See Barber's answer).

On the other hand, Van der Waerden's theorem for the coloring by two colors (which easily implies the general case of every finite coloring) can be derived by this statement without much effort (**Edit.** See the comments). So I think of it as an equivalent form of original Van der Waerden theorem.

I had conjectured a stronger version which I was unable to prove or disprove it:

Conjecture: If $W$ is an infinite word over finite alphabet. Then for every positive integer $k$, there exists $k$ disjoint and consecutive intervals in $W$, with the same number of occurrence of each letter.

I also know that a weaker version of this conjecture for binary alphabet and "the same" replaced by "proportional" is correct.

Does there exist a similar known result? Can someone give a prove or counterexample?