A $\Delta^0_2$-poset with no positive presentation

S. Yu. Podzorov in [Mat. Trudy, 9, No. 2, 109–132 (2006)] proved the validity of the following
THEOREM. If $\langle L,\le_L\rangle$ is a local lattice and $v$ a numbering of $L$ such that the relation $v(x)\le_L v(y)$ is $\Delta^0_2$-computable, then there is a numbering $\mu$ of $L$ such that the relation $\mu(x)\le_L\mu(y)$ is computably enumerable.
Podzorov also asked whether the hypothesis that $\langle L,\le_L\rangle$ is a local lattice is needed or the theorem is true of any partially ordered set (poset). We answer his question by constructing a poset for which the theorem fails.