Conjectures on dual-unitary involutive R-matrices and pair creation

Let $R$ be a unitary involutive $R$-matrix. An invertible matrix $\alpha$ is assumed to satisfy the relation referred to as Eq. (Ralpha_def-graphical), and $\theta$ denotes the partial traces of $R$; $\mathcal{D}'[R]$ is the associated space or set appearing in the source. Dual-unitarity and partial-trace conjectures. (1) If there exists an invertible $\alpha$ satisfying the stated relation, then $R$ must be dual unitary. (2) If $R$ is dual unitary, then

$$\theta\equiv \operatorname{Tr}_1[R]=\operatorname{Tr}_2[R]\in \mathcal{D}'[R]$$

and

$$\theta^2=\mathds{1}.$$

These are presented as conjectures concerning unitary involutive $R$-matrices that admit pair creation. The supplied text gives no resolution status, so both assertions remain open.