Not an answer, just an illustration for $6$ points, $0 \le x+y \le 2$.
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[![Poly6][1]][1]
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 - $A,B,C$: Point $3$ cannot connect to $1$ or $6$, so it must connect to
   $2,5$ or $4,5$ or $2,4$.
 - $B$: Point $6$ is now isolated by $34$ from $1$ and $2$.
 - $C$: Point $1$ is now isolated by $34$ from $5$ and $6$.
 - $D$: $1$ is trapped.
 - $E$: $2$ is trapped.


  [1]: https://i.sstatic.net/MGzql.jpg