Choice-independence of higher specialization for mixed Tate motives

Let $D(X,J)$ be the higher deformation associated with a simple normal-crossings stratum $Y_J$ and let $D^0(X,J)$ be its open deformation. Successive specialization along the $\mathbb{A}^1$ factors yields a functor on mixed Tate motives, while specialization along the diagonal yields the corresponding diagonal construction. Choice-independence conjecture. The successive specialization procedure does not depend on the choices when restricted to mixed Tate motive categories, and it agrees with the construction using the diagonal. This would establish that the higher specialization formalism is canonical in the mixed Tate setting, although the source gives no resolution of the claim.