It is hard to tell with such a poor reference. Twitter is not a scientific publication so statements there are not held to the usual scientific standards.
The Big Bang model is a classical (not quantum) model where the universe is modeled as being approximately isotropic and homogenous at the largest scales, and it was hotter and more dense earlier than it is now, and it has expanded and cooled as time progressed. There is ample observational evidence supporting this model.
If we extrapolate this model backwards in time, well past the point where we have observational data, then we get a singularity (what Carroll is calling the Big Bang "event"). Because this requires extrapolation beyond the point where we have observational data, it is reasonable to call this a conjecture.
There can be a hot dense universe that expands and cools without a singularity if our classical models break down sometime beyond the limit of our observational data. The details will require a quantum theory of gravity.
Since we know that in ordinary circumstances our classical models break down at small scales, many scientists (apparently including Sean Carroll) expect that this will happen for the Big Bang model as well. The result could be a solution to some as yet unknown cosmological equations that matches the Big Bang model at later times but diverges from it for the very dense early universe.