The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1 His example of a "long-standing conjecture" is the Riemann hypothesis, and he is cautioning "those who blithely assume the truth of the Riemann conjecture."
Q. What are examples of long-standing conjectures in analysis that turned out to be false?
Is Holt's adverb "often" justified?
1 Jim Holt. When Einstein Walked with Gödel: Excursions to the Edge of Thought. Farrar, Straus and Giroux, 2018. pp.36-50. (NYTimes Review.)