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Try:

const.inverse_matrix = target.matrix_world.inverted()   
obj.update_tag({'OBJECT'})

This does not show up immediatly. You have to update the view.

Try:

const.inverse_matrix = target.matrix_world.inverted()   
obj.update_tag({'OBJECT'})
bpy.context.scene.update()

This does not show up immediatly. You have to update the view / tag the region for redraw.

Try:

const.inverse_matrix = target.matrix_world.inverted()

This does not show up immediatly. You have to update the constraint somehow.

scene = bpy.context.scene
scene.frame_set(scene.frame_current)

Try:

const.inverse_matrix = target.matrix_world.inverted()   
obj.update_tag({'OBJECT'})

This does not show up immediatly. You have to update the view.

Try:

const.inverse_matrix = target.matrix_world.inverted()

Try:

const.inverse_matrix = target.matrix_world.inverted()

This does not show up immediatly. You have to update the constraint somehow.

scene = bpy.context.scene
scene.frame_set(scene.frame_current)