Rankin--Selberg integrals of opposite conductor--one newforms

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $Π_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $Π_{n+1}\times