I believed that until I understood the twinning process. Logically under your belief there would have to be an infinite number of souls, since an infinite number of twins can be split from the fertilized egg. Or, souls would have to split, which to me would be even stranger. There is a point a few days (or so) later when it becomes impossible to further split to twins, which makes more logical sense as a point in time when one single soul could be present. Something happens to the fertilized egg at that point which is still a scientific mystery. I believe that's a more likely point in time for the instant a soul enters a body.
With God, all things are possible. I believe that there are an infinite number of souls available.