Elizabeth Woodville & Desmond

Speaking of truth/lies how much truth is there in the story that Woodville signed
Desmonds death warrant - unbeknown to Edward - the result being that not only was he
executed but his two small sons - the youngest one asking the executioner to take care as
he had a boil on his neck. Is it true? Is it lie?

This was (supposedly) because she had it in for him because Edward had asked his
opionion of the (new) Queen and Desmond had answered something on lines of although
the queen was beautiful he should have married someone of higher rank - Edward then,
foolishly (maybe misjudging true character of his new wife) informed her of the
conversation.

I have come across this story several times but Im confused as to how true it is if at all? If
you know what I mean?

Eileen