Let $\mathcal{F}=(X,f,B)$ and $\mathcal{F}^{\prime}=(X^{\prime},f^{\prime},B^{\prime})$ be smooth proper Lagrangian fibrations inducing integral affine structures $\mathscr{A}$ and $\mathscr{A}^{\prime}$ on $B$ and $B^{\prime}$ respectively. Assume there exist Lagrangian sections $\sigma$ and $\sigma^{\prime}$ of $f$ and $f^{\prime}$ respectively. Then an integral affine isomorphism $\phi$ between $B$ and $B^{\prime}$ induces a symplectic $(\psi,\phi)$-conjugation between $\mathcal{F}$ and $\mathcal{F}^{\prime}$ such that $\psi\circ\sigma=\sigma^{\prime}\circ\phi$.