What is wrong is the idea that one can actually make the disk rotate; and it will remain perfectly rigid.

In reality, what this correct argument shows is that [relativity doesn't admit the existence of any perfectly rigid bodies][1]. Be sure that despite all the confused users' negative votes, this is a perfectly basic, settled, and indisputable textbook material that every mature physicist knows. The first sentence of this paragraph contains a link to the Gravity Probe B website.

When one takes a solid disk and makes it rotate, it will do all kinds of things resulting from the "imperfection of the material". It will tear apart by the centrifugal force, and if it won't, it will either tear basically along radial lines, or it will bend (the disk won't be planar anymore) because the circumference really shrinks by the Lorentz factor. If there existed a material that is perfectly rigid and cannot stretch or bend or tear, then it would be impossible to make it spin.

However, the non-existence of such a material may be shown even microscopically. It is not possible to "order" any solid object to keep the proper distances at every moment because the distance between two atoms (or points on the solid object) may only be measured with a delay $\Delta t = \Delta x / c$ simply because no information may move faster than light. That's why it's always possible to squeeze <em>any</em> rod on one end and the opposite end of the rod won't move at least for this $\Delta t = \Delta x / c$. 

In fact, the delay will be much larger than that, dictated basically by the speed of sound, not by the speed of light. Whatever material you have, relativity guarantees that it can be squeezed as well as stretched as well as bent.

  [1]: https://einstein.stanford.edu/content/relativity/q2018.html