Equality-frequency conjecture for Birch–Erdős representation functions

Let $f_{p,q}(n)$ be the Birch–Erdős representation function. Equality-frequency conjecture.

$$f_{p,q}(n+1)=f_{p,q}(n)$$

for infinitely many $n$. This conjecture accompanies the proposed sign-change phenomenon for $q>p>2$ and is intended to describe further fluctuations of the function; no resolution is given in the source.