Computability of distributive lattices

The class of (not necessarily distributive) countable lattices is HKSS-universal, and it is also known that the class of countable linear orders is not universal with respect to degree spectra neither to computable categoricity. We investigate the intermediate class of distributive lattices and construct a distributive lattice with degree spectrum $\{\mathbf d\colon\mathbf d\neq\mathbf0\}$. It is not known whether a linear order with this property exists. We show that there is a computably categorical distributive lattice that is not relatively $\Delta^0_2$-categorical. It is well known that no linear order can have this property. The question of the universality of countable distributive lattices remains open.