My guess is more prosaic. I'm thinking a private investigator hired a couple of low-budget thugs to snatch the gauze so that a paternity test could be performed. Afterwards, the so-called relic will be "returned" anonymously to local police amid sighs of relief and thanksgiving. Shortly thereafter no one will even wonder any longer about who stole it or why.
But a time bomb has been set ticking--"the slipping of small stones" and all that. The results of this paternity test will not be made public any time soon. It was performed solely to satisfy the conscience of the individual who requested it. When and if the revelations do come to light, it will be at precisely the time foreordained by God to deliver yet another blow to the conciliar sect, which may be many decades hence.