Injectivity conjecture for the positive quantum affine subalgebra of the $q$-tetrahedron algebra

Let ${\mathcal A}_q$ be the unital associative algebra generated by $x,y$ subject to the two cubic $q$-Serre relations

$$x^3y-[3]_q x^2yx+[3]_q xyx^2-yx^3=0,$$ $$y^3x-[3]_q y^2xy+[3]_q yxy^2-xy^3=0.$$

For each $i\in\mathbb Z_4$, let the homomorphism from ${\mathcal A}_q$ to $\boxtimes_q$ send the standard generators to $x_{i,i+1}$ and $x_{i+2,i+3}$, respectively. Injectivity conjecture. The map in Proposition is an injection.

The algebra ${\mathcal A}_q$ is often called the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$. The conjecture would realize this positive quantum affine subalgebra inside the $q$-tetrahedron algebra; the supplied text gives no resolution status.