I tried an example where control pts are :
{0,0,1,1}, {1,2,1,1},{2.5,0,1,h3},{4,2,1,1},{5,0,1,1}
knot vectors of [0 0 0 1 2 3 3 3] with third order bspline curve problem
Now when h3 =0 the third coord for the control p.t would be {0 ,0 ,0 ,0} and it works!
Note if h3 =0.25 say the control p.t would {0.625, 0, 0, 0.25, 0.25} which agrees with
what you stated earlier. Are you surprised?