The statement that "as $p \rightarrow \infty$, the minimum of the objective function is reached" is not correct. In fact, it is a pretty meaningless statement.
Commonly, a QAOA circuit has $p$ layers and $2p$ parameters. For a QAOA circuit to sample a low energy state with high probability, we need to find $2p$ parameters which are optimal or close to being optimal. For an arbitrary problem and a finite $p$, finding the optimal $2p$ parameters is already really hard. Non-convex optimization is hard.
This is to say that if we have infinitely many layers and all infinitely many parameters are suboptimal, then there is no reason to expect any sort of convergence to the ground eigenstate. A simple example would be a $p$-layered circuit with all parameters set to zero. Then we can set $p$ to any number, but we'll never reach the ground eigenstate. This is because the unitary matrix representing QAOA would be simply an identity matrix. So $p \rightarrow \infty$ does not imply an optimal solution will be found.
Having cleared this up, the correct statement is a quote from Farhi and Goldstone paper:
The algorithm depends on an integer p ≥ 1 and the quality of the approximation improves as p is increased.
The statement is a bit vague, but if you are familiar with Farhi's work, it is straightforward to see that the statement has to do with an approximation of the adiabatic evolution given by a time-dependent Hamiltonian. You can read Section 6 of the original paper for more details.
Roughly speaking, QAOA is viewed as a discretized version of an adiabatic evolution where we transition from the "simple" Hamiltonian, whose ground state is known, to a more "difficult" Hamiltonian, whose ground state we would like to find.
Putting this in the context of combinatorial optimization means that the simple Hamiltonian is the "mixer" and the "difficult" Hamiltonian is the Hamiltonian encoding a combinatorial problem. To implement the adiabatic process, we discretize it (finite time intervals + Trotterization) and represent it as a quantum circuit. Specifically, splitting the evolution time into $p$ subintervals and performing the first-order Trotterization produces a circuit with $p$ layers. As $p$ increases the "quality" of the approximation improves.
The paper that you referenced is nothing more than discretizing an adiabatic evolution given by different time-dependent Hamiltonians. In the QAOA case, we discretize the adiabatic evolution given by the transverse-field Ising Hamiltonian. In the QAO paper case, we discretize some other suitable Hamiltonians which we think can do a better job on some specific problem class.
So if you look one level up to where it all stems from (adiabatic algorithms), the QAOA and QAO are essentially two cases of a more general single theoretical paradigm. Hence, all convergence guarantees stem from the adiabatic theorem and not particular circuit implementation.
