Definition: A "$k$-chain" is a multi-graph obtained from a path of length $k$ by duplicating every edge.
Note that the number of paths between two endpoints of a $k$-chain is $2^k.$
Question: Let $G$ be a simple graph on $n$ nodes and let $s$ and $t$ be two nodes of $G.$ Suppose that number of (simple) paths from $s$ to $t$ in $G$ is at least $n^k.$ Then, is it possible to obtain a $\Omega(k)$-chain from $G$ with $s$ and $t$ as endpoints by a sequence of deletion and contraction of edges?
I would be equally happy with $\Omega(\sqrt k)$-chain or $\Omega(k^\alpha)$ for any $\alpha > 0.$
This question is closely related to another one that I asked few days ago: Do graphs with large number of cycles always contain large necklace minor?
I would appreciate any partial answer or any intuition on whether such a conjecture should hold.