The shuffle-of-areas closure conjecture

Let $\operatorname{rcl}$ be the right-closure operator and let $\operatorname{rot}$ be the rotation operator on words. For letters $i_1,\dots,i_{2k}$, define the area factors

$$\mathtt{i}_{2r-1}\mathtt{i}_{2r}-\mathtt{i}_{2r}\mathtt{i}_{2r-1},\qquad r=1,\dots,k.$$

Shuffle-of-areas closure conjecture. If all $i_j$ are distinct, then

$$(\mathtt{i}_1\mathtt{i}_2-\mathtt{i}_2\mathtt{i}_1)\shuffle\cdots\shuffle(\mathtt{i}_{2k-1}\mathtt{i}_{2k}-\mathtt{i}_{2k}\mathtt{i}_{2k-1})$$

is not in $\operatorname{im}(\operatorname{rcl}\circ\operatorname{rot})$, whereas it is contained in that image when the letters are not all distinct. Furthermore, for every $u\in\operatorname{im}(\operatorname{rcl}\circ\operatorname{rot})$,

$$(\mathtt{ij}-\mathtt{ji})\shuffle u\in\operatorname{im}(\operatorname{rcl}\circ\operatorname{rot}).$$

The claim refines the preceding computations on shuffles of area elements and describes when such shuffles yield letter-reduced conjugation invariants. The source gives examples showing both inclusion and non-inclusion, but the general assertion remains conjectural.