Unwrapped continuation invariance in Lagrangian Floer theory: energy and $C^0$ estimates

We consider pairs of Lagrangian submanifolds $(L_0,L),(L_1,L)$ belonging to the class of Lagrangian submanifolds with \emph{conic} ends on \emph{Weinstein manifolds}. The main purpose of the present paper is to define a canonical chain map $h_{\text{CL}}:CF(L_0,L)\to CF(L_1,L)$ of Lagrangian Floer complex inducing an isomorphism in homology, under the Hamiltonian isotopy $\CL={L_s}_{0 \leq s\leq 1}$ generated by \emph{conic} Hamiltonian functions such that the intersections $L\cap L_s$ do not escape to infinity. The main ingredients of the proof is an a priori bound for general isotopy of the energy \emph{quadratic} at infinity and a $C^0$-bound for the \emph{$C^1$-small} isotopy $\text{CL}=\{L_s\}$, for the associated pseudo-holomorphic map equations with \emph{moving} Lagrangian boundary induced by a conic Hamiltonian isotopy. For the Lagrangian submanifolds with \emph{asymptotically conic} ends, we construct a natural homomorphism $h_{\text{LL}}:HF(L_0,L)\to HF(L_1,L)$ for which the corresponding chain map may \emph{not} necessarily exist. This provides a more conventional construction of the chain isomorphism which replaces the sophisticated method using the Lagrangian cobordism via the machinery of \cite{kasturi-oh1,kasturi-oh2} whose details were only outlined in \cite{oh:gokova}.