False Dmitry I keeps the throne

False Dmitry I was the most successful of the Russian royal impostors, and managed to be crowned czar after fooling people into believing he was Ivan the Terrible's son. His marriage to Polish noblewoman Marina Mniszech angered Russians after she refused to convert from Catholicism to Orthodoxy, and his reign lasted less than a year.

Is there a way for False Dmitry I to survive a little longer and start a successful dynasty of his own? Maybe convincing Marina Mniszech to become Orthodox would salvage his reign?

(What little I know about this period comes from Wikipedia articles and half-remembered readings of A Treasury of Deception, so I'm probably wrong about something important.)
 
Getting Marina to become Orthodox would help. It would at least demonstrate independence from Poland. The only gripe would come from legitimate Rurikids, and the OTL claimant (Mikhail I) might have been satisfied with an important position in the Orthodox Church.

Of course, this butterflies away the Romanov dynasty which affects everything from the sale of Alaska to the Bolshevik revolution to the world wars.
 
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