Existence of a space with disjoint tightness but without ODT

Determine whether there exists a topological space that has disjoint tightness—meaning that for every subset A ⊆ X and every point x that is an accumulation point of A, there exist subsets A1, A2 ⊆ A with A1 ∩ A2 = ∅ such that x is an accumulation point of both A1 and A2—but does not have open disjoint tightness (ODT), where ODT requires that for every open set U ⊆ X and every point x ∈ U there exist disjoint open sets U1, U2 ⊆ U such that x is an accumulation point of both U1 and U2.

Background

The paper introduces open disjoint tightness (ODT) and relates it to disjoint tightness, showing that for submaximal spaces the two notions are equivalent. It also notes known separations in the other direction: there exists a countable regular space having ODT but not disjoint tightness. The authors ask whether the reverse separation is possible—i.e., whether disjoint tightness can occur without ODT—and provide examples of spaces without ODT (e.g., maximal spaces and certain compactifications), but do not resolve the existence question.