The bounded-arity symmetric-part conjecture for lattice varieties

Let $\mathcal{V}$ be a lattice variety axiomatized by identities of arity at most $k$. Let $\mathcal{V}_k^{sym}$ denote the variety axiomatized by the symmetric identities of arity at most $k$ that are true in $\mathcal{V}$.

Bounded-arity symmetric-part conjecture. For any lattice variety axiomatized by identities of arity at most $k$, one has

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

This is presented as a stronger variant of the conjecture that every lattice variety equals its symmetric part. The source does not state whether the bounded-arity claim has been resolved.