Well-definedness and invariance conjecture for Legendrian embedded contact homology

Let $(Y,\Lambda)$ be a pair consisting of a closed contact $3$-manifold $Y$, a closed Legendrian link $\Lambda \subset Y$, and a contact form $\alpha$ that is non-degenerate for $(Y,\Lambda)$. For $L$-admissible choices of data $\delta$ and $\delta'$, write $ECH^L(Y,\Lambda;\delta)$ for the filtered Legendrian embedded contact homology constructed from those choices. Legendrian embedded contact homology conjecture. The following hold: for any two $L$-admissible choices, there is a natural isomorphism

$$ECH^L(Y,\Lambda;\delta) \simeq ECH^L(Y,\Lambda;\delta');$$

for any $K>L$, there is a filtration map

$$\iota^K_L:ECH(Y,\Lambda) \to ECH^K(Y,\Lambda), \qquad \iota^M_L=\iota^M_K\circ\iota^K_L;$$

and the colimit

$$ECH(Y,\Lambda):=\underset{L}{\operatorname{colim}} ECH^L(Y,\Lambda)$$

depends only on $(Y,\Lambda)$ up to contactomorphism of the pair. The claim proposes that the construction is independent of admissible auxiliary data, has compatible filtration maps, and yields a contactomorphism invariant after taking the colimit; its resolution is not established in the supplied text.