No optimal planar integral point set with a unit edge for at least six points

Let $\mathfrak{M}(2,n)$ denote planar integral point sets with $n$ points, and call $M$ optimal when it attains the minimum possible diameter among sets in $\mathfrak{M}(2,n)$. No-unit-edge optimality conjecture. Every set $M\in\mathfrak{M}(2,n)$ with $n\geq 6$ such that $|M_1M_2|=1$ for some $M_1,M_2\in M$ is not optimal. The claim extends the observed existence of optimal sets containing distance $1$ for $n=3,4,5$; the source gives higher-point facher optimal sets as motivation, but leaves the assertion open.