No. A subdivision that can be lifted to a convex cap is called regular (or coherent, or weighted Delaunay). Here is an example of a non-regular subdivision:
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/aGq2d.jpg

For more on this, I recommend the book "Triangulations" by De Loera, Santos, Rambau.