Were Richard III's parents second cousins as expected?

Isabella of Castille:
 Unfaithful wife of Edmund,
Duke of York?

As I wrote previously, the investigation of Richard III's skeleton revealed that he does not share the same Y-DNA as his putative paternal-line relative Henry Somerset, 5th Duke of Beaufort and there are 18 possible birth events where non-paternity may have occurred.

There are four possible breaks between Edward III and Richard's own birth. One of the more probable false paternity events would be the birth of his grandfather, Richard of Conisburgh. Another is the birth of John of Gaunt. Provided the researchers were able to extract sufficient autosomal DNA, there is a simple, free, test which may reveal that there is a false paternity within these five steps of the broken line (John of Gaunt-Edward III-Edmund of York-Richard of Conisburgh-Richard Plantagenet-Richard III).

Gedmatch.com has a tool which can take autosomal DNA testing results and compare one side of a chromosome pair against the other, looking for similarities in regions which are expected to vary. It can then estimate which percentage of the DNA on each side is likely to have come from a common ancestor and how many generations back (basically based on amounts shared). Everyone gets one side of each chromosome pair from their father and one from their mother and if the parents are not related, there will be the expected variety and little similarity. The test evaluates how related an individual's parents are without having to test anyone but the individual.

According to the genealogy, Richard's parents are supposed to be, at their closest relationship, 2nd cousins (generations to most recent common ancestor would be about 4). His maternal grandmother's father should be the brother of his paternal grandfather's father. These brothers are John of Gaunt and Edmund of York and both paternity events are key interconnected steps on the broken line.

Additionally, Richard's father is supposed to be the 2nd cousin twice removed of his mother as his father's mother, Ann Mortimer, was also descended from Edward III, as the granddaughter of one of his granddaughters. Finally, his parents should have an additional more distant cousinship from common ancestry that is not related to Edward III.

Therefore, five steps in the paternity line happen to be uniquely tied to this supposed 2nd cousin relationship. Running Richard III's autosomal DNA results through the parental consanguinity calculator should indicate his parents have a common ancestor about 4 generations back. If the result is that the ancestor is farther back than 4 generations (or that his parents are unrelated), there is a break in the paternity line in the five steps between John of Gaunt and Richard III.

The key question is, have the researchers recovered enough autosomal DNA?