Is any mathematican more famous for their conjecture(s) than their theorem(s)?

Lothar Collatz: while he is a celebrated mathematician who has a formula named after him (Collatz-Wielandt formula) and he has received quite a few honorary degrees (see wiki), he is definitely best known for his Collatz conjecture, also known as the $3n+1$ conjecture, which he posed in 1937.

It remains unsolved up to this day, despite numerous attempt by professional and amateur mathematicians, and its popularity can be seen even here. In fact, I would argue that the best "proof" that his conjecture is more famous than his actual work, is that the tag (collatz) refers exclusively to the Collatz conjecture.

Andrew Beal, while not strictly a mathematician, has not proven (to my knowledge) any mathematical result, despite having a rather famous unsolved problem in his name with a monetary prize that is the same of that of any Clay Mathematics prize.

Franz Mertens was a late 19th/early 20th century German mathematician who is known for some results about the density of prime numbers and even has a constant named after him. But what he is probably most famous for, no doubt because of the elementary nature of the statement, is the Mertens conjecture, which stated:

Mertens' conjecture: For all $n>1,$ $$\left\lvert \sum_{k=1}^{n}\mu(k) \right\rvert < \sqrt{n},$$ where $\mu\colon\mathbb{N}\to\{-1,0,1\}$ is the Möbius function.

This conjecture, if true, would imply the Riemann hypothesis (!) - hence its fame. Unfortunately, in spite of tremendous amounts of numerical evidence in favour, it was proved false in the 1980s. The smallest counterexample is known to be larger than $10^{14},$ but the upper bound on it is truly enormous: $\operatorname{exp}{(1.59\times10^{40})}.$