Postnikov-tower mixed Hodge complex conjecture for perfect complexes

Let $T$ be constructed as a Postnikov tower

$$T=T_n\to\dots\to T_i\to T_{i-1}\to\dots\to F_2\to T_0=*$$

and suppose that each stage has a fibration

$$\kappa(V_i,i)\longrightarrow T_i\xrightarrow{\tau_i}T_{i-1}.$$

Here $\mathcal{U}_i$ denotes the proalgebraic group associated with the cohomologies of the trivial principal bundle, and $\mathfrak{u}_i$ is its Lie algebra.

Postnikov mixed Hodge complex conjecture. For each $i$ there exists a unipotent proalgebraic group $\mathcal{U}_i$ with a mixed Hodge structure, and the groups $\mathcal{U}_i$ form a mixed Hodge complex with maps

$$\mathfrak{u}_i\otimes\mathfrak{u}_j\longrightarrow\mathfrak{u}_{i+j-1}$$

coming from the Whitehead product on $T$.

This is proposed as a local generalization for mapping spaces into perfect complexes, where the nonabelian part of the mixed Hodge structure is expected to be trivial. The source does not state whether this conjecture has been proved or disproved.