It may "generally" be true, but "generally" doesn't feed the bulldog.
I have a Sigma f/2.8 24-60 that isn't as sharp wide open at 55mm as either my 18-55 kit lens or my el-cheapo 55-200 VR.
Set them all at f/8 and it is a pick-em for sharpness, they are all about the same.
The vaunted Nikon 70-200 VR f/2.8 is not significantly sharper wide open at either 70mm or at 200mm than my 55-200mm VR... the big difference is that wide open, it is 2.8 as opposed to the 4-5.6 of the 55-200. At f/11, they are virtually identical in sharpness.
Obviously, the f/2.8 is vastly superior for shooting things like sports and other events where light is limited, there is absolutely no comparison in build quality, the VR on the 70-200 is MUCH better, it focuses more quickly and accurately, yadda yadda yadda... it also his better when it comes to things like CA, etc... but... for simple sharpness at f/11? There is virtually no difference.
Lets take it one step further... you really have to know what you are doing to shoot an f/2.8 wide open with group shots... You do get different (and much smaller) depth of field with fast glass wide open than you do with slow glass. When I first got my 70-200 (my first f/2.8 lens) I thought there was something wrong with it, because it wasn't very sharp... turns out it was operator error (not surprisingly, Nikon does actually check to make sure $1700 lenses can actually focus before they ship them out), I was taking picture of subjects where the depth of field was causing problems... Completely different shooting technique is required, and I am learning that. Shot CORRECTLY, focused CORRECTLY, the 70-200 is quite sharp wide open... but then again, so is my lowly little el-cheapo 55-200 VR.