Let $t$ be a lattice term. Define $t^\circ$ by replacing every $\land$ with $\lor$ in $t$ and vice-versa. In the variety of distributive lattices, the "majority term" $$ M(x, y, z) := (x \land y) \lor (x \land z) \lor (y \land z) $$ is self-dual in the sense that $M \approx M^\circ$ is an identity of distributive lattices. However, one can verify that any NON-distributive lattice will actually NOT satisfy this identity! (So this is actually an equivalent characterization of distributive lattices.)
Further, one can verify, using the explicit solution to the word problem for lattices, that if $t \approx t^\circ$ is an identity of lattices, then $t \approx x$ for some atomic term $x$. (I wrote a blog post in which I do all of this verifying if you want to see the details.)
So, the natural question is this: what is the situation for the variety of modular lattices? According to the nlab, free modular lattices have an undecidable word problem, so the proof strategy for lattices fails. But $M_3 \vDash M \neq M^\circ$, so we can't just plug in the majority term either.
One strategy I did have was the following: let $M_n$ be the lattice with only a top, a bottom, and $n$ atoms, say $c_i$ for $i \in n$. Let $x \mapsto x^*$ be the evident anti-automorphism of $M_n$, i.e. it swaps the top and bottom and fixes everything else. It is useful to observe that given any such anti-automorphism, $t^\circ(\vec{x}) = t(\vec{x}^{ *})^*$. Let $t[\vec{x}]$ be an $n$-arity term. If $t \approx t^\circ$ is an identity of modular lattices, we have $$ M_n \vDash t(\vec{c}) = t^\circ(\vec{c}) = t(\vec{c}^{\ *})^* = t(\vec{c})^* $$ with the first equality because $M_n$ is modular, the second equality because of the previous observation, and the third equality because $c_i^* = c_i$ for all $i \in n$. So we can conclude $$ M_n \vDash t(\vec{c}) = c_i $$ for some $i \in n$. This tells us that $t$ cannot be symmetric, unlike $M$, so if a solution does exist, it likely won't be nice. But this lack of niceness could perhaps be used to prove non-existence. I'm not very familiar with modular lattices, so it's hard for me to go beyond this.
