Would holocaust casualties have been reduced if WAllies emphasized it?

I was wondering if holocaust casualties could have been reduced if the allies had made the plight of the victims a core component of their propaganda around 1941. Perhaps Churchill and/or FDR could have made a major speech denouncing/exposing the holocaust and promising retribution to its perpetrators? Also, leaflets could be dropped over German and German occupied territory emphasizing how to hide potential victims and the consequences of collaboration with the Nazis?

It seems that the Nazis tried to cover up the holocaust when they realized they were losing. Perhaps if they realized the world knew then they’d at least not construct the extermination camps.

I was thinking the Hungarian Jews and other communities in a similar situation might survive the war if the allies follow the above ideas? I don’t know if it was too late for ex. The Polish Jews.
 
The Nazis wouldn’t have stopped. They would have denied everything and continued exactly as before. Genocide was that central to their ideology.
 
IIRC the Nazis blamed international jewry for the American/British “conspiracy”against them. Enphasizing the Holocaust would only be seen as confirmation of their paranoid beliefs.
 
Hitler believed the English were part of his Aryan race and believed they would thank him for what he was planning to do through the HOLOCAUST. Just speeches would have been seen as propaganda.
 
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