The specialization conjecture for symmetry-breaking link homology

Let $L$ be a link, let $E^{\ast,\ast}_2(L,\overline{\theta})$ denote the $E_2$-page associated to the specialized twist $\overline{\theta}$, and let $\wp(x)=x^n$ be the algebraic specialization. The groups $E^{\ast,\ast}_q(L,\overline{\theta})$ denote the pages of the resulting spectral sequence.

Specialization conjecture. Under the algebraic specialization $\wp(x)=x^n$, the groups

$$E^{\ast,\ast}_2(L,\overline{\theta})$$

are isomorphic to $sl(n)$-link homology. This specialization has a topological lift, meaning that it can be lifted to a map of filtered spectra. Consequently, there is a spectral sequence

$$E^{\ast,\ast}_q(L,\overline{\theta}),\qquad q\geq 2.$$

The conjecture is motivated by computations for the Hopf link and the $(2,3)$-torus knot. The proposed topological lift is intended to provide a route toward resolving the conjecture.