The symmetric-part conjecture for lattice varieties

Let $\mathcal{V}$ be a lattice variety. Its symmetric part $\mathcal{V}^{sym}$ is the variety axiomatized by all symmetric lattice identities true in $\mathcal{V}$, where a lattice identity is symmetric when both of its terms are invariant under every permutation of its variables.

Symmetric-part conjecture. Each lattice variety $\mathcal{V}$ equals its symmetric part:

$$\mathcal{V}=\mathcal{V}^{sym}.$$

The conjecture asks whether all lattice varieties are determined by their symmetric identities. Its status is not specified in the source.