The adjective mendacious
refers to something that is not truthful; or that it is lying or false. You can certainly fix a falsehood (or a mendacity) by telling or revealing the truth — and be willing to accept the consequences if you're the one who started the lie.
In Tess of the d'Urbervilles, by Thomas Hardy, John Durbeyfield is walking home. He encounters a local parson who tells him of his family history: The poverty-stricken Durbeyfields are descended from the once famous d'Urbervilles, a wealthy family dating back to the time of William the Conqueror.
Here, John questions the parson about the location of the family manor:
And where do we raise our smoke, now, parson, if I may make so bold; I mean, where do we d'Urbervilles live?"
"You don't live anywhere. You are extinct — as a county family."
"Yes — what the mendacious family chronicles call extinct in the male line — that is, gone down — gone under."
"Then where do we lie?"
"At Kingsbere-sub-Greenhill [cemetery]: rows and rows of you in your vaults, with your effigies under Purbeck-marble canopies."