Is a uniformly random $r$-regular bipartite graph $r$-edge connected with high probability?

A graph is $$r$$-edge connected if the number of edges in a minimum cut is at least $$r$$.

It is known that a random $$r$$-regular graph is $$r$$- vertex connected (which implies $$r$$-edge connected) with high probability as the number of vertices grows. See Introduction To Random Graphs, Frieze & Karonski, pg. 214 for a proof.

Can the same be said for a uniformly random $$r$$-regular bipartite graph?

The configuration model for random bipartite graphs is actually slightly simpler. In the case of random $$r$$-regular bipartite graphs on $$2n$$ vertices:
1. Take $$2nr$$ points partitioned into $$2n$$ cells $$A_1 \cup \dots \cup A_n \cup B_1 \cup \dots \cup B_n$$, each of size $$r$$.
2. Pick a uniformly random bijection $$f : A_1 \cup \dots \cup A_n \to B_1 \cup \dots \cup B_n$$.
3. Define a graph with vertices $$a_1, \dots, a_n, b_1, \dots, b_n$$; for each time $$f(x) = y$$ with $$x \in A_i$$ and $$y \in B_j$$, add an edge $$a_i b_j$$ to the graph.
This is not guaranteed to be a simple graph: we might have parallel edges. However, with positive probability (for constant $$r$$) it is a simple graph; conditioned on being simple, any $$r$$-regular bipartite graph on $$2n$$ vertices is equally likely to be the result.