The conjectured formula for the first minimum zero eigenvalue multiplicity

Let $m_1^0(n,w)$ denote the minimum multiplicity of the first zero eigenvalue in the relevant Johnson-scheme setting, with parameters $n$ and $w$. The preceding results reduce its determination to comparing explicit candidate values, and computational evidence motivates the following claim.

Conjectured identity. For $w\geq 5$ and $n\geq 2w+1$,

$$m_1^0(n,w)=2{n-2 \choose w-1}.$$

The identity is supported by computations for $6\leq n\leq 600$ and $3\leq w\leq n/2$; the paper presents it as a conjecture, so its general validity remains open.