Example of Divergence of a Greedy Algorithm with Respect to an Asymmetric Dictionary

We construct an example of an asymmetric dictionary $D$ in a Hilbert space $H$ such that the linear combinations of elements of $D$ with positive coefficients are dense in $H$, but the greedy algorithm with respect to $D$, in which inner product with the elements of $D$ (not the modulus of this inner product) is maximized at each step, diverges for some initial element.