Let $T\to A$, $T\to B$ be homomorphisms of (finitely generated) $k$-algebras where $k$ is an algebraically closed field of characteristic $0$. Assume moreover that $R$ is another $k$-algebra (reduced, finitely generated) with a homomorphism of $k$-algebras $\varphi:R\to A\otimes_T B$.
Assume that for every $m_1\in\mathrm{Specm}(A)$, $m_2\in\mathrm{Specm}(B)$, $n_1\in\mathrm{Spec}(A\otimes_T(B/m_2))$ and $n_2\in\mathrm{Spec}((A/m_1)\otimes_TB)$ the natural maps $\varphi_{m_1}:R\to (A/m_1)\otimes_T B$ and $\varphi_{m_2}:R\to A\otimes_T(B/m_2)$ satisfy that $\varphi_{m_1}^{-1}(n_2)\to n_2/n_2^2$ and $\varphi_{m_2}^{-1}(n_1)\to n_1/n_1^{2}$ are surjective.
Question: Is it true that for every $m\in\mathrm{Specm}(A\otimes_T B)$, the map $\varphi^{-1}(m)\to m/m^2$ is surjective?
This has to do with my (unanswered) question Embedding of fiber products of schemes