Dan Romik’s answer excellently addresses your title question. I think it’s also elaborating on a couple of aspects of how the context you describe affects it.
Most importantly: I think the scenario you sound worried about — where the professor and/or colleagues solve the conjecture quickly and write it up, without your involvement — is extremely unlikely. If that happened, it would be quite unethical of the professor. If a conjecture has been publicly announced, then it’s certainly generally considered fair game for anyone to work on. But if an interesting conjecture is communicated to you privately, especially by a student or a junior colleague, then it would be — at least — pretty bad form to work on it further without keeping the person who suggested it involved.
What I would be doing in your professor’s situation — and I think most academics would do something roughly along the lines — is something like the following: First of all, I would try to figure out how hard the conjectures are. Do I know them, or do I easily see that they follow from other results I know? (Presumably not, based on the reply you received.) If not, ask around some colleagues in case they recognise the conjectures, or see a clear relationships between them and known techniques/results; and also think a bit harder about them myself, and perhaps do a bit of literature search. (It sounds like your prof. is at this stage.) Depending on what I can find/figure out at this stage, then:
If someone recognises the conjecture as known, or as an obvious consequence of existing results (where “obvious” means roughly “if you show someone the results and the conjecture side by side, it’s easy to see that the conjecture follows from the results”), or, conversely, if the conjectures are false for similarly known reasons: Then I’d write back to the student to tell them, and congratulate them on (re)discovering an interesting fact, and suggest keeping in touch about research project possibilities in the future (depending on what kind of projects the department’s programme offers). In this case, we’ve all had a fun problem to solve together, but nothing is publishable. This is honestly the most likely scenario — not just for a student suggestion, but for most questions anyone comes up with. That’s just how research is!
If it looks like the conjectures are not obvious but are approachable using techniques within the student’s reach/background: I would suggest that the student works on this as a research project, under my or a colleague’s supervision. (Again, this will depend partly on how “student research projects” fit into the department’s programme/curriculum.) I’d aim to stay fairly hands-off, and giving no more guidance as the student needed. If this goes well, it could well be publishable, with the student as first or sole author.
If it looks like the conjecture is best approached using deeper theoretical tools, beyond what the student can be expected to master in a short time-frame, but is reasonably approachable using those tools, then I might work on it myself or with colleagues, but certainly also keeping the student involved in the discussions (both to introduce them to the techniques involved, and give them the opportunity of contributing if they get up to speed enough on the techniques). This might well result in a paper, probably including the student as an author. (If the conjecture was interesting enough to spark such a project, then it’s most likely enough of a contribution to merit authorship.)
If I and colleagues can’t easily see how to approach the question at all, I’d congratulate the student on finding an interesting and difficult problem. I’d keep it at the back of my mind, and if I later have an idea on how to approach it, I’d proceed as in case (2) or (3). If it’s interesting enough, I might also mention it to colleagues further afield, and would mention that I got it from a student. This is the only case that could reasonably lead to a solution without the student as co-author: if researchers one or two degrees removed from you hear the conjecture, see a solution, and write it up. Hopefully, I would find out (directly or via the grapevine) that they had solved it, in which case I would suggest they at least acknowledge you by name, and possibly invite you as a co-author. In this case, the criteria from Dan Romik’s answer for whether you deserve co-authorship or just acknowledgement would apply.
In all cases: if the conjectures are interesting and novel enough that a solution could be publishable, I would certainly make sure to keep the student in the loop about anything subsequently done with them; and I think most academics would consider it unethical if the professor didn’t do so.