Conjecture on the motive isomorphism between a symmetric Kloosterman motive and a rank-three motive

Let $\mathrm{M}_2^6$ and $\mathrm{M}_3^{(2,2)}$ be the motives introduced in the paper, and let $(-3)$ denote the Tate twist. Motive-isomorphism conjecture. The two motives

$$\mathrm{M}_2^6(-3)\quad\text{and}\quad\mathrm{M}_3^{(2,2)}$$

are isomorphic. The source notes that their $\ell$-adic realizations are already isomorphic and presents the motive-level isomorphism as the conjectural strengthening; no resolution is given.