Nonexistence conjecture for diameter perfect codes in Johnson spaces

Let $J(n,w)$ denote the Johnson space of binary words of length $n$ and weight $w$, equipped with the Johnson distance. A diameter perfect code is a code attaining equality in the code-anticode bound; the relevant known examples are Steiner systems and their complements.

Diameter-perfect-code conjecture. There are no nontrivial diameter perfect codes in $J(n,w)$, except for Steiner systems and their complements.

The conjecture is motivated by the analogous conjecture for perfect codes and by results showing that the existence of such a diameter perfect code implies the existence of Steiner systems. The source reports no proof or disproof, so the conjecture remains open.