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Just an illustration of @RichardKlitzing's construction in 3D:
His $n+1 = 4$ layers are: $$ v_1 \;,\; \{v_2,v_4,v_5\} \;,\; \{v_3,v_8,v_6\} \;,\; v_7 $$
Just an illustration of @RichardKlitzing's construction @RichardKlitzing's construction in 3D: