An example of a disjunctive 0,1-simple lattice which is neither 0-simple nor 1-simple.

Question

This is a follow-up to my previous question: A 0-simple lattice which is not a 0,1-simple lattice.. Following the terminology of the previous question, I define a disjunctive 0,1-simple lattice to be a lattice $L$ with more than one element such that for every non-constant homomorphism $f$ of $L$, either $f$ preserves the bottom element of $L$, or $f$ preserves the top element of $L$. (But some homomorphisms may preserve the top but not the bottom, or vice versa). Is there an example of such a lattice which is neither 0-simple nor 1-simple?