Just infinite quotients of finitely generated subgroups of $PL^+[0,1]$

We show that just infinite quotients of finitely generated subgroups of Richard Thompson's group F are virtually abelian, answering a question of Grigorchuk. We show the same holds for the group of piecewise linear orientation preserving homeomorphisms of the interval, and the group of piecewise projective homeomorphisms of the real line. The latter provides a plethora of examples of nonamenable groups with this property.