Embedding median graphs into minimal distributive ∨-semi-lattices

It is known that a distributive lattice is a median graph, and that a distributive ∨-semi-lattice can be thought of as a median graph i every triple of elements such that the inmum of each couple of its elements exists, has an inmum. Since a lattice without its bottom element is obviously a ∨-semi-lattice, using the FCA formalism, we investigate the following problem: Given a semi-lattice L obtained from a lattice by deletion of the bottom element, is there a minimum distributive ∨-semi-lattice L d such that L can be order embedded into L d ? We give a negative answer to this question by providing a counter example.