Given the false claims of this lineage research:
http://washington.ancestryregister.com/FALAISELineage00006.h... ,
historian and gossip mongers aren't to be considered reliable evaluators of fact where wishful thinking is concerned.
At any rate, supposing that your own line of Carey Y DNA inheritance didn't have an interruption by a non-Carey male or an adoption in 800 years of many generations, should they ever discover and run a DNA analysis of King Henry's remains you could easily come tantalizing close to verifying that gossip by comparing your own DNA with King Henry's results. As King Henry suffered from debilitating illnesses brought about by an exceedingly meat heavy diet, a DNA analysis might also prove useful in modern times medically.
Such DNA results would not be absolute evidence, but with the other circumstantial evidence surrounding the case it more than enough to convict beyond a reasonable doubt.