The independence number conjecture for even Johnson-type graphs $J_{\pm}(n,4,1)$

Let $J_{\pm}(n,4,1)$ be the Johnson-type graph with parameters $n$, $4$, and $1$, and let $\alpha[G]$ denote the independence number of a graph $G$. Independence number conjecture. Let $n>n_0$ be an even number. Then

$$\alpha[J_{\pm}(n,4,1)]=2n(n-2).$$

This is posed as an open question about determining independence numbers of Johnson-type graphs; the source does not provide a proof or resolution for this parameter range.