Rank conjecture for symmetric \mathfrak{gl}_1-homology and reduced triply graded homology

Let $L$ be an uncolored link. Its symmetric $\mathfrak{gl}_1$-homology and reduced triply graded homology are graded homology theories associated to $L$, and the total rank of either homology means the rank of the underlying total homology group after forgetting the gradings.

Rank conjecture. The total rank of the symmetric $\mathfrak{gl}_1$-homology is equal to the total rank of the reduced triply graded homology.

This conjecture concerns the relationship between the symmetric $\mathfrak{gl}_1$-homology and Khovanov–Rozansky's triply graded homology. Spectral sequences from the triply graded homology to the symmetric $\mathfrak{gl}_n$-homologies are known to exist, but the corresponding symmetric case is presented here as a conjecture.