Here is a sketch of their arguments:

• •

  Using the asymptotic assumption on the exact Lagrangian $L$, one can assign an analytic invariant $A(L)$ to $L$ as follows. Let $f_{L}:L\to\mathbb{R}$ be a primitive of the Liouville form $\lambda$, namely $df_{L}=\lambda|_{L}$, then $f_{L}$ converges to constants $c_{0},c_{\phi}$ at the two asymptotic ends along $\Pi_{0},\Pi_{\phi}$ respectively. Then one defines $A(L)=c_{\phi}-c_{0}$. If $L$ coincides with the Lawlor neck $L_{\phi,A}$, then $A(L)=A$.

• •

  Partially compactify $\mathbb{C}^{n}$ into a Liouville manifold identified as the plumbing $M$ of two cotangent bundles $T^{*}S^{n}$ with $T^{*}S^{n}$. Here the two copies of $S^{n}$ arise topologically as one-point compactifications of $\Pi_{0}$ and $\Pi_{\phi}$ by adding the points at infinity $\infty_{0}$ and $\infty_{\phi}$, and topologically $M$ is the union of $\mathbb{C}^{n}$ and the two cotangent fibres over $\infty_{0}$ and $\infty_{\phi}$ respectively. Under suitably fast decay condition at infinity, the unknown special Lagrangian $L$ can be compactified into an exact graded embedded Lagrangian $\bar{L}$ inside $M$. One would like to compare this to the Lagrangian $\bar{L}_{\phi,A}$ obtained by the compactification of the standard Lawlor necks $L_{\phi,A}$ inside $M$.

• •

  By analyzing the intersection pattern with the two cotangent fibres at infinity, and using the classification results of Abouzaid and Smith [1], one shows that inside $D^{b}Fuk(M)$, the Lagrangian object $\bar{L}$ is isomorphic to one of the two Lagrangian connected sums of the two $S^{n}$ with suitable gradings, and in fact the assumption on Floer degrees $\sum\phi_{i}=\pi$ singles out $\bar{L}\simeq\bar{L}_{\phi,A}\in D^{b}Fuk(M)$, the opposite surgery corresponding to $\sum\phi_{i}=(n-1)\pi$. This step needs $n\geq 3$. For contradiction, we assume $\bar{L}$ does not coincide with $\bar{L}_{\phi,A}$ for any choice of parameter $A>0$.

• •

  By a modification of the Thomas-Yau argument, one shows that after a small Hamiltation perturbation $\bar{L}^{\prime\prime}$ of $\bar{L}_{\phi,A}$, we can ensure $\bar{L}^{\prime\prime}$ is transverse to $\bar{L}$, there is no degree $0,n$ intersection points in $\bar{L}^{\prime\prime}\cap\bar{L}$ inside $\mathbb{C}^{n}$, and there is precisely one intersection point $p$ and $q$ in $\bar{L}^{\prime\prime}\cap\bar{L}$ on each of the two cotangent fibres at infinity respectively. Morever, the $D^{b}Fuk(M)$ class and the analytic invariants of $\bar{L}^{\prime\prime}$ agree with that of $\bar{L}_{\phi,A}$.

  Remark 2.8.

  The subtlety at infinity prevents one from removing degree $0,n$ intersections outside the $\mathbb{C}^{n}$ region, so one does not reach an immediate contradiction as in the Thomas-Yau argument. This technical failure is necessary, because the Lawlor necks with fixed asymptotic planes are not unique, but do arise in a 1-parameter family. It is in overcoming this technical problem that holomorphic curves appear in [40].

• •

  Now suppose the Lawlor neck is chosen with the parameter $A=A(L)$, which presumes $A(L)>0$.

  Lemma 2.7.

  [40, Thm 2.15] Assume $J$ is a generic almost complex structure on $M$ compatible with the Liouville structure. There exists a $J$-holomorphic strip $\Sigma$ with boundary on $\bar{L}$ and $\bar{L}^{\prime\prime}$ and two corners at $p$ and $q$ respectively.

  Proof.

  Consider the Floer cup product with mod 2 coefficients

  $$HF^{0}(\bar{L}^{\prime\prime},\bar{L})\otimes HF^{0}(\bar{L},\bar{L}^{\prime\prime})\to HF^{0}(\bar{L},\bar{L}),$$

  which can be identified as the cup product

  $$H^{0}(\bar{L},\bar{L})\otimes H^{0}(\bar{L},\bar{L})\to H^{0}(\bar{L},\bar{L}),\quad 1_{\bar{L}}\cup 1_{\bar{L}}=1_{\bar{L}},$$

  and thus must be nontrivial. However, at chain level this Floer product comes from the $A_{\infty}$ operation

  $$m_{2}:CF^{0}(\bar{L}^{\prime\prime},\bar{L})\otimes CF^{0}(\bar{L},\bar{L}^{\prime\prime})\to CF^{0}(\bar{L},\bar{L}),$$

  which must be nontrivial. The counting interpretation implies there are intersection points $p^{\prime}\in CF^{0}(\bar{L},\bar{L}^{\prime\prime})$ and $q^{\prime}\in CF^{0}(\bar{L}^{\prime\prime},\bar{L})\simeq CF^{n}(\bar{L},\bar{L}^{\prime\prime})^{\vee}$ and some holomorphic strip in between. Since degree $0,n$ intersection points cannot occur inside $\mathbb{C}^{n}$, they can only occur at infinity, so we must have $\{p,q\}=\{p^{\prime},q^{\prime}\}$. ∎

  Now the area of the $J$-holomorphic curve can be computed cohomologically. Using the choice of parameter $A$,

  $$\int_{\Sigma}\omega=\int_{\partial\Sigma}\lambda=\int_{p^{\prime}\to q^{\prime}}df_{\bar{L}}+\int_{q^{\prime}\to p^{\prime}}df_{\bar{L}^{\prime\prime}}=\pm(A(L)-A(L^{\prime\prime}))=\pm(A-A)=0.$$

  This contradicts the positivity of area of the holomorphic curve, which proves $L$ must coincide with $L_{\phi,A}$.

• •

  Finally one needs to a priori justify $A(L)>0$. This relies on a slightly more complicated holomorphic polygon counting argument, and the main upshot is that one can produce a nontrivial holomorphic triangle from a distinguished triangle in $D^{b}Fuk(M)$, with the three edges on $\bar{L}$, $\Pi_{0}\cup\{\infty_{0}\}$ and $\Pi_{\phi}\cup\{\infty_{\phi}\}$. Then one shows $A(L)$ has the interpretation as its area, so must be positive.