If one digs around in the chart, this Richard is "closer*" to the House of York and House of Tudor than the House of Plantagenet, acknowledging that "Plantagenet" was antecedent to both:
Dame Elizabeth Woodville-Grey = married secondly Edward IV (York)
Firstly = Sir John Grey of Groby
Their son, Thomas Grey KG 1st Marquess of Dorset = Cecily Bonville
Their son, Hon Thomas Grey KG KB PC 2nd Marquess = Margaret Wotton of Kent-Boughton Malherbe
Their son, Henry Grey 3rd Marquess of Dorset = Lady Frances Brandon Duchess of Suffolk
Their daughter (2 of 3+), Lady Katherine Grey Lady Herbert Countess of Hertford was Richard's ancestor
However; their daughter (1 of 3+) and Katherine's sister, Lady Jane Grey Nine Days Queen, House of Tudor
Of course, part of this "analysis" presumes that *Royal Blood flows back in time.
Perhaps another House of Plantagenet connection exists?