The algebra conjecture for parameterizing motivic local systems

Let $Y$ be a smooth projective geometrically connected variety over the finite field $\mathbb F_q$, let $D$ be an effective divisor on $Y$, and let $N$ be a positive integer. Let $\mathcal C_{\mathbb F_q}$ be the tensor category used in the source, and let $\phi_n$ denote its numerical realization over $\mathbb Q$.

Algebra conjecture for motivic local systems. There exists a commutative associative unital algebra $A=A_{Y,D,N}$ in $\mathcal C_{\mathbb F_q}$ such that, for every $n\ge 1$, $\phi_n(A)$ is semisimple over $\mathbb Q$, and for every prime $l$ with $(l,q)=1$ there is a bijection

$$\operatorname{Hom}_{\mathbb Q\text{-alg}}(\phi_n(A),\overline{\mathbb Q})$$

with the elements of $\operatorname{IrrRep}^{\mathrm{mot,geom}}_{Y\times_{\mathbb F_q}\mathbb F_{q^n},N}$ whose ramification divisor is $D$. The bijection is equivariant for the natural action of ${\operatorname{Gal}}(\overline{\mathbb Q}/\mathbb Q)\times\mathbb Z/n\mathbb Z$.

This conjecture proposes a finite-dimensional algebra whose geometric points parameterize motivic local systems with prescribed ramification. It would organize the expected independence from $l$ and the Galois and cyclic symmetries, but is not established in the source.