The correspondence conjecture for transverse-link and branched-cover contact invariants

Let $L$ be a transverse representative of an alternating smooth link. Write $\psi(L)\in\widetilde{Kh}(L)$ for the transverse-link invariant in reduced Khovanov homology, let $\xi_L$ be the contact structure on the branched double cover $\Sigma(L)$, and let $c(\xi_L)\in\widehat{HF}(-\Sigma(L))$ be its Heegaard Floer contact invariant. Let $c_0(\xi_L)$ denote the image of $c(\xi_L)$ in the corresponding subquotient of the associated graded group, namely $c_0(\xi_L)\in\widehat{HF}_0(-\Sigma(L))/\widehat{HF}_1(-\Sigma(L))$. The Ozsváth–Szabó spectral sequence identifies the reduced Khovanov homology of $L$ with the associated graded group of $\widehat{HF}(-\Sigma(L))$. Correspondence conjecture. If $L$ is a transverse representative of an alternating smooth link, then the homological grading of $\psi(L)\in\widetilde{Kh}(L)$ is the same as the filtration level of $c(\xi_L)\in\widehat{HF}(-\Sigma(L))$, and this filtration level is $0$. Moreover, under the isomorphism between $\widetilde{Kh}(L)$ and the associated graded group of $\widehat{HF}(-\Sigma(L))$, one has \$\psi(L)=c_0(\xi_L)\\$. This conjecture proposes a precise relationship between the transverse Khovanov invariant and the Heegaard Floer contact invariant; the stated grading and identification remain the conjectural content here.