How hard is it to succeed in pure math research after an adviser's death midway through PhD?

TL;DR: How hard would it be to finish out a Ph.D. and have an effective early career as a pure mathematician after my adviser has unexpectedly passed away? My department does not have anyone else able to take on students in my field at the time. I am currently in my fourth year.

I read this similar question on academia.stackexchange, but it seemed to address practical plans for going forward more than the costs and benefits of doing so.

I am a pure mathematics Ph.D. student in my fourth year. My adviser recently passed away unexpectedly, and my department does not have anyone else in his field who is able to take on students.

In the next six months or so, my department has offered me the option of working with a couple of postdocs at my current institution, as well as flying out to occasionally meet with more senior mathematicians at different institutions. It is unclear whether the department will hire another researcher in my field anytime soon, especially one who is not fresh out of a postdoc.

I currently have enough material for roughly one paper, but I feel that I still need guidance to finish my Ph.D., and I felt like I was really benefiting from weekly/biweekly meetings with a single adviser. My adviser helped keep me confident and excited, as well as explaining difficult techniques and offering new approaches to proofs or counterexamples.

I have heard some people advise on transferring institutions at this point, though my department does not recommend that. In addition, some mathematicians/postdocs have pointed out to me that an adviser is an important early-career research collaborator, to the extent that it is difficult to do good research early in a career without having them to collaborate with or ask questions of.

If my time to Ph.D. is going to be lengthened significantly, I am not certain I want to continue. So my question is: how hard would it be to finish out a Ph.D. and have an effective early career as a pure mathematician without an adviser in the traditional sense?