There wouldn't be any such study because it presumes that there is such a thing as "true" and "false" history.

"History" (as in, a written narrative) is at best an approximation of what actually occurred in the past. It is necessarily abstracted and simplified, and it is necessarily incomplete. We talk about "discovering" the past as if it were some entity that exists in a single form outside of the present, but the truth of the matter is that history only exists in the present.

That isn't to say that we can't come to agree on a historical consensus about what did or did not happen based on available evidence. There are some things we can definitely agree happened (such as the nuclear bombing of Japan) based on the huge preponderance of evidence. We can also agree on some things that are very unlikely to have happened, for instance, space travel prior to the 20th century based on the complete lack of evidence.

However, history isn't just about "facts" or easily confirmed events, it is largely an exercise in analyzing those events and what underlies them. For instance, what led to the decision to bomb Japan in WWII. Answering that kind of historical question is much less quantifiably "true" or "false" simply by looking at the available evidence. Even if you had access to all of the available evidence for the event, two different historians might come to different conclusions about what led up to the bombing. Now, odds are no one is going to say anything as outlandish as that it was really a plot by Martian's controlling the US government, but there can certainly be some differences in the two different accounts written by different historians, even using the same evidence.

The question then is how you are going to quantify what is or is not true in a historical narrative, and then convert that into a statistical model.