The weight-support system conjecture for perfect codes containing Preparata codes

Let $C$ be a binary perfect code containing a Preparata code. For an extended perfect code, consider the codewords of weight $4$; for a non-extended perfect code, consider the codewords of weight $3$. Their supports form the corresponding Steiner system. A system is linear when it is equivalent to the system formed by the supports of the weight-$3$ words of the linear Hamming code.

Weight-support system conjecture. All codewords of weight $3$ of a perfect code containing a Preparata code, and all codewords of weight $4$ of an extended perfect code containing a Preparata code, form a linear Steiner triple system and, respectively, a linear Steiner quadruple system.

The conjecture is based on the common structure of the two known perfect codes containing Preparata codes. The paper does not establish it in general.