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.