I didn �t see the exact sentence (as you wrote it above) in the article.
 
 Serious doubts about claims for fusion in collapsing bubbles--
someone is claiming (saying) that they can achieve fusion in collapsing bubbles--others have some serious doubts about it (they don �t believe it is true/actually works).
 
make a mistaken claim - a mistaken claim  (saying they can do something, but they can �t really do it--they are mistaken) 
 
in any  branch of science- it doesn �t matter what you are researching, if others seriously doubt you (or prove you wrong) that branch becomes a pariah (very unpopular and nobody wants anything to do with it).
 
Overall, the sentence as written does not really make much sense, perhpas this is better:
A mistaken claim in any branch of science, for example the serious doubts about claims of fusion in collapsing bubbles, make...