Recently, I collected a short list of phrases for conjectures on a well-known social media platform and several people contributed to it. One can easily find more examples online, but I like my list, so I will keep it here and include references (as far as I have them). Probably, I will add more entries over time.

Firstly, I will give a list of phrases. Secondly, references for the phrases.

Phrases for Conjectures

1. Conjecture that something is true.
2. Ask if something is true.
3. Say that it is tempting to conjecture something.
4. Say that it is tempting to risk the conjecture that something is true.
5. Say that it is natural to wonder if something is true.
6. Have a slide with a chicken conjecturing something.
7. Say that something is not impossible.
8. Say that something seems safe to conjecture.
9. Make your PhD student conjecture something in the PhD thesis.
10. Your PhD student conjectures something on its own in the PhD thesis.
11. Take a previous conjecture and rephrase it, so that it conjectures something new, but claim that it is the old conjecture.
12. Write a second paper where you reference your first, conjecture-less paper and claim that you made your conjecture in the first paper.
13. Intentionally, conjecture the opposite of something that you believe to be true.
14. State it as a theorem and claim the proof does not fit in the margin.

References

1. While this seems to be the obvious way to state a conjecture, it is actually not that easy to find examples for it. Indeed, my list phrases lengthened while I was looking for a more famous conjecture which was explicitly stated as a conjecture. Here is my first example: The abc conjecture was suggested by Masser and Osterlé. Masser called it a conjecture. After I was done finding a very famous example, I realized that I actually know another example from my area. Cameron and Liebler made a conjecture in “Tactical Decompositions and Orbits of Projective Groups” about what I call Boolean degree 1 functions of vector spaces in this post:
2. I am very sure that I saw this, but I failed at finding an example for now. As soon as I find one, I will add it here. Edit (11/09/2022): Igor Pak suggested the following example from “Brunn-Minkowski inequalities for contingency tables and integer flows” by Alexander Barvinok:
3. This is in this list due to Bill Martin misremembering the next entry, but it is in use (as Google showed me), see for example “On sums and products of integers” by Paul Erdős and Endre Szemerédi.
4. Suggested by Bill Martin. This is famously used by Philippe Delsarte in his PhD thesis which you can find here. More precisely, he writes: “It is tempting to conjecture that there are no nontrivial perfect codes in the Johnson graphs.”
5. Kelly’s formulation of the Reconstruction Conjecture.
6. Here I lack a reference and I am not 100% sure if this colleague of mine wants to be mentioned here. The colleague claimed to be too much of a chicken to conjecture something, so a chicken on a slide of her talk makes a conjecture.
7. Paul Erdős phrased the Erdős Matching Conjecture that way in “A problem on independent r-tuples”.
8. Willem Haemers and Edwin van Dam phrase their conjecture that almost all graphs are determined by their spectrum like this here.
9. This and the next two are both inspired from Ryser’s conjecture. There are several conjectures known as Ryser’s conjecture, see this article by Best and Wanless on this confusion. The beginning summarizes this sufficiently:
10. See one above.
11. See one above.
12.  This is discussed in the previous post. Frankl and Rödl write in this paper from 1987 that this paper by Frankl from 1986 makes a conjecture on the independence number of the orthogonality graph, even so nothing in that direction seems to be indicated in the 1986 paper. To be fair, Frankl conjectured something very similar in 1986 (which is not comparable to the conjecture above):
13. This is not really a phrase, but it still seems good advice by a colleague of mine. He argues that it motivates people if they can “disprove” you. Again, I lack a written reference for this claim. It is more of a meta-concept anyway. Edit (10/02/2020): Gil Kalai pointed out that he has this old post from 2008 on Shmuel Weinberger‘s take on why one conjectures things. It includes this point and five others.
14. This is of course a reference to Fermat’s conjecture. Suggested to me by John Bamberg from SymOmega.