Classification conjecture for tight symmetric inner distribution codes in Johnson schemes

Let $r,s$ be positive integers, let $n:=r+s$, and consider the Johnson scheme $J(2n,n)$. A tight symmetric inner distribution code is a code for which equality holds in the relevant bound for the Johnson scheme, and its degree is $s$. Classification conjecture. If there exists a tight symmetric inner distribution code for $J(2n,n)$ of degree $s$, then either

$$s=1$$

or

$$s=n-1.$$

The conjecture is motivated by computer experiments showing that the associated polynomial has all integral zeros only rarely, except in these two cases. No resolution is supplied in the source, so the classification remains open.