The combinatorial injection conjecture for dominance order on compositions

Let $A,a,B,b$ be integers with $A \geq B \geq 0$ and $a \geq b \geq 0$. For $m,n \in \mathbb{N}$, let $\mathcal{U}^m_n$ denote the set of compositions of $n$ with $m$ non-negative parts. For compositions $\lambda$ and $\mu$, write $\lambda\mu$ for their concatenation, and write $\trianglerighteq$ for the dominance order induced by the non-increasing rearrangements. Combinatorial injection conjecture. There is an injection

$$\gamma: \mathcal{U}^A_b \times \mathcal{U}^B_a \hookrightarrow \mathcal{U}^A_a \times \mathcal{U}^B_b$$

such that, for every $(\lambda_1,\mu_1) \in \mathcal{U}^A_b \times \mathcal{U}^B_a$, if $(\lambda_2,\mu_2)=\gamma((\lambda_1,\mu_1))$, then

$$\lambda_1\mu_1 \trianglerighteq \lambda_2\mu_2.$$

This conjecture gives a combinatorial formulation implying the case $k=2$ of the paper's theorem relating total nonnegativity of the matrices $\mathcal{T}_p$ and $\mathcal{S}_p$.