`eliminate` returns nothing?

The goal is to eliminate ,  and  from . However, eliminate only outputs a null expression (I added a {} to emphasize it): 

restart;expr := 4*[y*z/((x + y)*(x + z)), z*x/((y + z)*(y + x)), x*y/((z + x)*(z + y))]:{eliminate}([a, b, c]**~2 =~ expr, [x, y, z]);
 = {}

Why is the result empty? 
In my view, the result should be  (or its equivalent). One may verify this by: 

seq(seq(seq(
    is(eval((a^2 + b^2 + c^2 - 4)^2 = (a*b*c)^2, 
      elementwise([a, b, c] = [k1, k2, k3]*sqrt(expr)))), 
    `in`(k3, [-1, +1])), `in`(k2, [-1, +1])), `in`(k1, [-1, +1]));
 = true, true, true, true, true, true, true, true