Why does the transformation in the proof for SL=L preserve connectedness of s and t?
The proof of Theorem 4.1 states that:


  Let $S$ be the connected component of $G$, such that $s \in S$.
  By the arguments above, $S \times [N]$ is a connected component of $G_{\mathrm{reg}}$, and $G_{\mathrm{reg}}|_{S\times [N]}$ is non-bipartite. [...]
  
  By Lemma 3.2 and Lemma 3.3, we have that $\lambda(G_{\exp}|_{S\times[N]\times([D^{16}_{\mathbf{e}}])^\ell}) \leq 1/2$. In particular, we have that $S\times[N]\times([D^{16}_{\mathbf{e}}])^\ell$ is a connected component of $G_{\exp}$.


In particular, if $t \in S$ then both $s'$ and $t'$ belong to the connected component $S\times[N]\times([D^{16}_{\mathbf{e}}])^\ell$.

Lemma 3.2 only works when $G$ is non-bipartite and $\lambda(H) \leq 1/2$. Perhaps one of these conditions wasn't satisfied in your examples?