Algebraic Grünbaum–van Kampen–Flores conjecture

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

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

then

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

The conjecture is an algebraic analogue of classical non-embeddability results and would imply Grünbaum's theorem on joins of skeleta of simplices. Its resolution status is not specified in the source.