The lonely element conjecture for numerical semigroups

Let $S$ be a numerical semigroup, and call a nonzero small element of $S$ lonely if it has the property specified in the source. Lonely element conjecture. If every nonzero small element of $S$ is lonely, then $S$ is dimension-$2$ realizable, equivalently, its matricial dimension is at most $2$. The paper states that this conjecture is disproved by a counterexample; the converse is known from earlier work.