Before revision:  We prove this claim by showing that a special case of an intimately related decision problem SCMP - does there exist a measure with a non-mass-splitting transport cost and support size below prescribed bounds? - is NP-hard for all rational data.
After revision:  We prove this claim by showing that a special case of an intimately related decision problem SCMP - does there exist a measure with a non-mass-splitting transport cost and support size below prescribed bounds? -- is NP-hard for all rational data.