Counterexample.
$N=\{0,1,2,3,4,5,6\}$
$P=\{u,v,w,y,z\}$
$u=\{0,1,3,5\},\ v=\{0,2,4,6\},\ x=\{0,1,2\},\ y=\{0,3,4\},\ z=\{0,5,6\}$
$\alpha=\frac12(1_u+1_v),\ \beta=\frac13(1_x+1_y+1_z)$
bof
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