For any $n \geq 1$, let $\Sigma_n$ denote the closed orientable surface of genus n. In http://arxiv.org/abs/1202.6302, the authors showed that for any $n$, there is a degree two map $ f: S^1 \times \Sigma_n \to \#_n S^1 \times S^2$ (the $n$-fold connected sum).

Question: For any $n \geq 2$, is it possible to produce a possibly different closed orientable surface $B$ together with a degree one map $f': S^1 \times B \to \#_n S^1 \times S^2 $ ?

For $n=1$ of course it is possible to (for example) set $B=S^2$ and find a degree one map (the identity).

For any $n \geq 1$, let $\Sigma_n$ denote the closed orientable surface of genus n. In http://arxiv.org/abs/1202.6302, the authors showed that for any $n$, there is a degree two, $\pi_1$-surjective, map $ f: S^1 \times \Sigma_n \to \#_n S^1 \times S^2$ (the $n$-fold connected sum).

Question: For any $n \geq 2$, is it possible to produce a possibly different closed orientable surface $B$ together with a degree one map $f': S^1 \times B \to \#_n S^1 \times S^2 $ ?

For $n=1$ of course it is possible to (for example) set $B=S^2$ and find a degree one map (the identity).