Conjecture on monochromatic Gower sum and product subspaces

Let $X_k$ be the underlying set of the Gower subspaces, and let $X_k=\bigcup_{i=1}^r C_i$ be a finite partition. A Gower sum subspace is generated by a subset $A=\{\mathbf{n_i}:i\in I\}\subset X_k$, while a product subspace is generated by a subset $B=\{\mathbf{m_i}:i\in I\}\subset X_k$. Monochromatic Gower subspace conjecture. For every $k,r\in\mathbb{N}$ and every such partition, there exist subsets $A$ and $B$ such that the Gower sum subspace generated by $A$ and the product subspace generated by $B$ are both contained in the same color class $C_i$. This would unify the additive and multiplicative Ramsey phenomena established separately in the paper, but the combined monochromatic conclusion is presented as a conjecture and its resolution is not supplied here.