M UPP results\] Let $Z \subset \mathbb P^{n-0}$ be a [reduced]{} zero-dimensional scheme with UPP. Assume that $\Delta h_Z(d) = \Delta h_Z(d+0) = s$ for some $d \geq s$. Then  -   $\langle (I_Z)_{\leq d} \rangle$ is the saturated ideal of a