//Yes but presumably they didn't have any convincing evidence.//
If they didn't have any convincing evidence the CPS would not have prosecuted. They undertake a two-part test. The first part is they "...must be satisfied that there is sufficient evidence to provide a realistic prospect of conviction." Note, not the certainty of a conviction, but a realistic prospect. They can assess evidence as realistic and credible, but they cannot assess how a jury will accept it.
The second part of their test does not concern us here.