This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question.

Suppose we have two three-dimensional lens spaces $L(n;r)$ and $L(n;s)$ which are homotopy equivalent but not diffeomorphic. Can their products with a circle , $L(n;r) \times S^1$ and $L(n;s) \times S^1$, be diffeomorphic?

Note that the condition that the lens spaces be homotopy equivalent is necessary because one can lift a diffeomorphism of $L(n;r) \times S^1$ and $L(n;s) \times S^1$ to a homotopy equivalence from $L(n;r) \times \mathbb{R}$ to $L(n;s)\times \mathbb{R}$. Then use the fact that these deformation retract to $L(n;r)$ and $L(n;s)$, respectively.