Carlet's conjecture on the sum-freedom of the binary multiplicative inverse function

Let $n$ be a positive integer and let $f_{\text{inv}}:\mathbb F_{2^n}\to\mathbb F_{2^n}$ be defined by

$$f_{\text{inv}}(x)=\begin{cases}1/x&\text{if }x\ne 0,\\0&\text{if }x=0.\end{cases}$$

A function is $k$th order sum-free if the sum of its values is nonzero on every $k$-dimensional affine subspace of $\mathbb F_{2^n}$. Carlet's conjecture. For $3\le k\le n-3$, $f_{\text{inv}}$ is not $k$th order sum-free.

The conjecture concerns the values of $k$ for which the binary multiplicative inverse function has sum-freedom over $\mathbb F_{2^n}$. It is resolved in the supplied source: the paper confirms the conjecture when $n$ is not a prime, while the stated status evidence indicates that a positive solution would determine all remaining values of $k$.