Duality conjecture for critical-handle Legendrian homology

Let $W$ be a Liouville cobordism corresponding to attaching critical handles along a collection $\Lambda$ of Legendrian spheres, and let $L{\mathbb H}^{\text{Ho}}(\Lambda)_*$ and $L{\mathbb H}^{\text{Ho}+}(\Lambda)_*$ be the associated Legendrian contact homology groups. For a field $\mathfrak{k}$, write $(\cdot)^\vee$ for the linear dual. Duality conjecture. The following isomorphisms hold:

$$SH^{-*}(W,\partial^+W)\cong L{\mathbb H}^{\text{Ho}}(\Lambda)_*,$$ $$SH_{-*}(W,\partial^+W)\cong SH^*(W,\partial^-W)\cong (L{\mathbb H}^{\text{Ho}}(\Lambda)_*)^\vee.$$ $$SH^{-*}_{<0}(W,\partial^+W)\cong L{\mathbb H}^{\text{Ho}+}(\Lambda)_*,$$ $$SH_{-*}^{<0}(W,\partial^+W)\cong SH^*_{>0}(W,\partial^-W)\cong (L{\mathbb H}^{\text{Ho}+}(\Lambda)_*)^\vee.$$

This is presented as a consequence of the preceding surgery conjecture and is intended to express duality between symplectic homology of the reversed cobordism and the Legendrian homology groups. Its resolution is not supplied in the source.