The Tokareva–Shapovalov conjecture on balanced Boolean functions as bent derivatives

Let $f$ be a balanced Boolean function in an even number $n$ of variables, of degree at most $n/2-1$. Suppose that there is a nonzero vector $y$ such that

$$f(x)=f(x\oplus y)$$

for every vector $x$.

Tokareva–Shapovalov conjecture. The function $f$ is a derivative of a bent function.

The paper states that this conjecture is disproved by constructing balanced Boolean functions satisfying the stated conditions that are not derivatives of bent functions.