Variant of Euler's conjecture

For a fixed exponent $n\ge4$, can one find a single positive integer base $b$ such that $b^n$ can be represented as a sum of exactly 3 positive $n$th powers, and as a sum of $n-1$ positive $n$th powers, and as a sum of any number of positive $n$th powers in between?

More formally: for every integer $n\ge4$, does there exist a positive integer $b$ such that, for every integer $k$ satisfying $2<k<n$, there exist positive integers $a_1,\ldots,a_k<b$ such that

$ b^n=a_1^n+a_2^n+\cdots+a_k^n? $

I am interested in whether this statement is known and true, or false for some $n$, or perhaps related to known results on sums of like powers.