The function $s(n)$ is defined, for integers $n\ge2$, to be the sum of all the proper divisors of $n$, that is, all the divisors other than $n$ itself. E.g., $s(6)=1+2+3=6$.

The aliquot sequence of $n$ is the sequence $n,s(n),s(s(n)),s(s(s(n))),\dotsc$.

The Catalan-Dickson Conjecture holds that all aliquot sequences are bounded.

The Guy-Selfridge Conjecture holds that most sequences for even $n$ are unbounded.

Catalan's statement, dated 18 April 1888, can be found at https://www.aliquot.de/literatur/catalan.pdf It can also be found in Catalan's Mélanges Mathématiques, https://archive.org/details/mlangesmathm00catauoft/page/n7/mode/2up?view=theater page 240. A citation for Dickson is Theorems and tables on the sums of divisors of a number, Quart. J. Math., v. 44, 1913, pp. 264-296.

Guy and Selfridge have several papers on aliquot sequences. I'm not sure where their conjecture appears.

I thank Lola Thompson for reminding me of this pair of contradicting conjectrures, in a talk she gave at a recent conference.