TITLE: "Long-standing conjectures in analysis ... often turn out to be false"
QUESTION [41 upvotes]: 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.)

REPLY [3 votes]: Fuglede's conjecture was open for 30 years (1974-2004) only to be proven false by T. Tao for dimensions $d\geq 5$ with a counter-example  arising from a set with an exponential orthonormal basis (a spectrum) in a finite abelian group which does not tile by translation. Interestingly, the largest dimension for which the conjecture is open now is $d=2$.