Semisimple decomposition conjecture for saturated log motives

Let $\mathfrak{M}_{\mathrm{sat}}$ be the category of saturated log motives, let $E$ be a log Tate curve, and let $M$ be an object of $\mathfrak{M}_{\mathrm{sat}}$. Saturated log-motive decomposition conjecture. There are classical Grothendieck motives $C_r$ over $k$, equal to zero for almost all $r$, such that

$$M\cong\bigoplus_{r\geq 0}\operatorname{Sym}^r(H^1(E))\otimes C_r.$$

The conjecture proposes that saturated log motives are generated by symmetric powers of the log Tate motive together with classical motives; it is motivated by the preceding proposition and the log Tate conjecture, with no resolution given.