Atomic determination conjecture for order continuous functionals on projective Banach lattices

Let $X$ be a projective Banach lattice, and let an order continuous functional mean a continuous linear functional preserving limits of decreasing nets to zero in order. The atoms of $X$ are its minimal nonzero positive elements. Atomic determination conjecture. All order continuous functionals on $X$ are determined by its atoms. The statement appears after a question about Dedekind-complete projective Banach lattices, but the supplied text gives no further formulation or evidence concerning its status.