I think a counterexample can be obtained via simplicial spheres. It is known that in higher dimensions, most simplicial spheres do not belong to a polytope. The dual of the face lattice of such a $d$-dimensional non-polytopal simplicial sphere is a $d$-regular "simple" poset in your sense, and cannot belong to a $d$-polytope (as otherwise, the original poset would too).
M. Winter
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