Robust transitivity with robust singularities near a product map

Let $M$ and $N$ be closed manifolds of arbitrary finite dimension. Suppose that $M$ supports an expanding map $F$. Product-map conjecture. The product $M\times N$ supports a $C^1$ robustly transitive map displaying robust singularities that is $C^1$-close to $(F,\operatorname{Id})$. A partial affirmative answer is known for skew products, while many related questions about these maps and constructions remain open.