No, there have been countless Presidents Trophy Winners that went on to win nothing, Presidents Trophy winners lose in the playoffs more often than they go on to become Stanley Cup Champions.
As of this current decade only 1 Presidents Trophy winner has managed to win the cup, the 2013 Blackhawks.
In the last 20 years only 5 Presidents Trophy winners went on to win the cup..
99' Stars
01' Avalanche
02' Red Wings
08' Red Wings
13' Blackhawks
Running away with #1 overall in the league standings is nice and all but history shows it shouldn't mean that you should be favourites to win the cup because of it...of note you can ask the 2009 Sharks, 2011 & 2012 Canucks, the 2014 Bruins, the 2015 Rangers, 2016 & 2017 Capitals and 2018 Predators.