Here's a possible proof that the limiting distribution is unique under all three stochastic convergence types, but is the proof valid? Your input would be most welcome.
Step #1: "The limiting distribution is unique if the convergence is in distribution"
Proof: According to Wikipedia
(a) convergence in distribution is metrizable,
(b) A metrizable space is Hausdorff,
(c) A Hausdorff space implies the uniqueness of limits of sequences.
The result follows.
Step #2: "The limiting distribution is unique if the convergence is in probability/almost surely"
Proof: According to Wikipedia,
(a) convergence in probability implies convergence in distribution,
(b) almost sure convergence implies convergence in probability
Hence the result.