Conjecture on variations of mixed Hodge-Tate structures for multiple polylogarithms

Let $Li_{n_1,\ldots,n_k}(x_1,\ldots,x_k)$ be a multiple polylogarithm, and let ${\mathfrak L}_n(x_1,\ldots,x_n):=Li_{\underbrace{\scriptstyle 1,\ldots,1}_{\scriptstyle n\text{ times}}}(x_1,\ldots,x_n)$ denote a multiple logarithm. Consider the variations of mixed Hodge-Tate structures associated with these functions and their limit mixed Hodge-Tate structures at infinity. Conjecture on multiple polylogarithms. The variations of mixed Hodge-Tate structures related to any multiple polylogarithm can be produced as the variations of some limit mixed Hodge-Tate structures related to some suitable choice of multiple logarithm. This would extend the explicit constructions for multiple logarithms and the known treatment of weight-three multiple polylogarithms and double polylogarithms; the general construction remains open.