The torus-generator chain-map conjecture for convex toric domains

Let $X_\Omega$ be a smooth convex toric domain with boundary $Y$. Let $C_*^{T^3}$ be the chain complex freely generated over $\mathbb{Z}/2$ by torus generators, with differential given by rounding a corner and locally losing one edge labeled $h$. For a convex generator $\Lambda$, let $\tilde{\Lambda}$ be the torus generator obtained by attaching to $\Lambda$ a vertical segment from $(0,0)$ to its upper-left endpoint and a horizontal segment from $(0,0)$ to its lower-right endpoint, labeling the new segments $e$. The torus-generator chain-map conjecture. One can choose the contact form $\lambda$ on $Y$ specified in the perturbation lemma and a generic $\lambda$-compatible almost complex structure $J$ on $\mathbb{R}\times Y$ such that the linear map

$$ECC^L(Y,\lambda,0,J)\longrightarrow C_*^{T^3},$$

sending the admissible orbit set $\imath(\Lambda)$ to the torus generator $\tilde{\Lambda}$, is a chain map. This conjecture would give an improved version of the theorem for smooth convex toric domains by identifying the relevant ECH differential with the combinatorial torus-generator differential.