Let $\phi=(p\vee(q\wedge r))\wedge (r\vee(p\wedge q))$ and let $\chi = (p\wedge (r\vee (p\wedge q)))\vee (r\wedge (p\vee (q\wedge r)))$. For your De Morgan lattice ${\bf M}$ we have $\phi\models_{\bf M} \chi$, but we have $\phi\not\models_{\bf L} \chi$ for the De Morgan lattice: 


 <img src="https://i.sstatic.net/1cn4Z.png" width="300" />

The complementation is defined by the self-duality that fixes $p$ and $r$.

This is more complicated than Adam's example, but it has the property that it does not refer to the negation.<p> 

The justification is this: Let ${\bf L}^*$ be ${\bf L}$ minus its top and bottom. ${\bf L}^*$ is the splitting lattice for the $p$-modular law, which is $\phi\leq \chi$. Any lattice either satisfies the $p$-modular law, or has a copy of ${\bf L}^*$ as a sublattice. Your lattice ${\bf M}$ doesn't have a sublattice isomorphic to ${\bf L}^*$, so it satisfies the $p$-modular law. On the other hand, ${\bf L}$ does not satisfy the $p$-modular law, as you can see by assigning to the variables $p, q, r$ the values indicated in the figure.