Powerful tableau noncommutative upper-bound conjecture

Let $P$ be a $(\mathbf{3}+\mathbf{1})$-free poset, let $\lambda\vdash n$, and let $m_\lambda^P(\mathbf{u})$ be the relevant noncommutative elementary coefficient in $\mathcal{R}_P=\mathbb{Q}\langle u_1,u_2,\ldots,u_n\rangle/I_P$. For a tableau $T$, let $\operatorname{colword}(T)$ denote its column word, and let $\mathbf{u}_{\operatorname{colword}(T)}$ be the corresponding word. Powerful tableau noncommutative upper-bound conjecture. The element

$$\left(\sum_{T\in\operatorname{powSSYT}_P(\lambda)}\mathbf{u}_{\operatorname{colword}(T)}\right)-m_\lambda^P(\mathbf{u})$$

has a $\mathbf{u}$-positive expansion in $\mathcal{R}_P$. This is a noncommutative refinement of the proposed powerful-tableau upper bound.