Sum-freeness conjecture for the multiplicative inverse function

Let $n$ be an odd integer with $n\ge 7$, and let $f_{\text{inv}}:\mathbb F_{2^n}\to\mathbb F_{2^n}$ be defined by

$$f_{\text{inv}}(x)=x^{-1}\quad(x\ne 0),\qquad f_{\text{inv}}(0)=0.$$

For $n\ge 2$, define

$$\mathcal K_n=\\{1\le k\le n-1: f_{\text{inv}}\text{ is not }k\text{th order sum-free on }\mathbb F_{2^n}\\}.$$

Sum-freeness conjecture. For odd $n\ge 7$,

$$\mathcal K_n=\\{3,4,\dots,n-3\\}.$$

Equivalently, for every integer $k$ with $3\le k\le n-3$, the multiplicative inverse function is not $k$th order sum-free. The claim is known for even $n$, while it remains open for odd $n$.