\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
Let \(\li_t:L\times \RR\to X\) be a parameterization of the submanifold \(K\).
By reparameterization, it suffices to check that this is a Lagrangian submanifold at points \((p, 0)\).
The tangent space \(T_{(p, 0)}(L\times \RR)\) is spanned by vectors \(\{\partial_{x_1}, \ldots, \partial_{x_{n-1}}, \partial_t\}\).
Because \(L\) is a Lagrangian of the fiber, and the fiber is a symplectic submanifold,
\[\li_t^*\omega(\partial_{x_i}, \partial_{x_j})=0.\]
Since \(K\) was constructed via parallel transport, \((\li_t)_*\partial_t\in (\ker(\pi_*))^{\omega\bot}\). Since \(L\) is a Lagrangian of the fiber, \((\li_t)_*\partial_{x_i}\in \ker(\pi_*)\).
Therefore,
\begin{align*}\li_t^*\omega(\partial_{x_i}, \partial_{t})=0
\end{align*}