Nearby Lagrangian conjecture

Let $M$ be a closed smooth manifold of dimension $n$, and let $\pi\colon T^{*}M\to M$ be its cotangent bundle, equipped with the tautological $1$-form $\lambda$ (in local canonical coordinates $(q_{1},\dots,q_{n},p_{1},\dots,p_{n})$, $\lambda=\sum_{i=1}^{n}p_{i}\,dq_{i}$) and the symplectic form $\omega=d\lambda$.

Call a submanifold $L\subset T^{*}M$ Lagrangian if $\dim L=n$ and $\omega|_{L}=0$, closed if $L$ is compact without boundary, and exact if there exists $f\in C^{\infty}(L,\mathbb{R})$ with $\lambda|_{L}=df$.

Let $0_{M}=\{(q,0):q\in M\}\subset T^{*}M$ denote the zero section. Say that two submanifolds $L_{0},L_{1}\subset T^{*}M$ are Hamiltonian isotopic if there is a smooth family of compactly supported functions $H_{t}\colon T^{*}M\to\mathbb{R}$, $t\in[0,1]$, whose time-dependent Hamiltonian vector fields $X_{H_{t}}$, determined by

$$\omega(X_{H_{t}},\cdot)=dH_{t},$$

generate a flow $\varphi_{t}$ with $\varphi_{0}=\mathrm{id}_{T^{*}M}$ and $\varphi_{1}(L_{0})=L_{1}$.

Then for every closed exact Lagrangian submanifold $L\subset T^{*}M$, the submanifolds $L$ and $0_{M}$ are Hamiltonian isotopic.