The linear-components conjecture for perfect codes containing Preparata codes

Let $C$ be a binary perfect code containing a Preparata code, and let an $i$-component denote the corresponding component of $C$. The set $\{(x,x,|x|)\}$ is the model linear component, where $|x|$ denotes the parity of $x$.

Linear-components conjecture. Every $i$-component of $C$, for every $i$, is linear; equivalently, it is equivalent to

$$\{(x,x,|x|)\}.$$

This conjecture concerns the structure of components in perfect codes containing Preparata codes. The paper gives no proof or resolution of the assertion.