Higher-dimensional failure of semigroup decompositions for semi-algebraic sets

For each dimension $d$, let $\mathfrak{A}_d$ be the class of sets admitting the semigroup decomposition considered in the paper, and call a set semi-algebraic if it is defined by polynomial inequalities. Higher-dimensional semigroup decomposition conjecture. There exist semi-algebraic sets in dimension $d \geq 4$ which do not belong to $\mathfrak{A}_d$, but all semi-algebraic sets defined by three-variable polynomials do belong to $\mathfrak{A}_3$.

This conjecture proposes that the dimension-three theorem is the last possible general result of this form, while the supplied text gives no resolution of the higher-dimensional assertion.