Algebraic Menger conjecture

Let $K$ and $L$ be simplicial complexes, and let $\vartheta$ denote their van Kampen obstruction. Algebraic Menger conjecture. If

$$\vartheta(K)\neq 0\quad\text{and}\quad\vartheta(L)\neq 0,$$

then

$$\vartheta(K\times L)\neq 0.$$

This is presented as a second algebraic non-embeddability conjecture in the study of Smith classes and the van Kampen obstruction. Its resolution status is not specified in the source.