Cusick–Li–Stănică-type conjecture for perturbations of symmetric Boolean functions

Let $\sigma_{n,k}$ denote the elementary symmetric Boolean function of degree $k$ in $n$ variables, and let $X_1$ be the first coordinate Boolean function. A perturbation is balanced when it takes the values $0$ and $1$ equally often. The trivial cases are those with

$$n=2^{r}m+k-1,$$

where $r=\lfloor \log_2(k_s)\rfloor+1$ and $m$ is a positive integer.

Cusick–Li–Stănică-type conjecture. No perturbation of the form $\sigma_{n,k}+X_1$ is balanced except for the trivial cases above.

The conjecture proposes that the explicitly identified trivially balanced perturbations are the only balanced perturbations of this form, extending a conjecture of Cusick, Li and Stănică for elementary symmetric Boolean functions. Its status is not resolved in the supplied source.