Chen ranks conjecture for Kähler groups

Let $X$ be a compact Kähler manifold. For each $g\geq 2$, let $\mathcal{E}(X)$ be the set of relevant fibrations $f\colon X\to B_f$ and define

$$h_g(X)=\#\big\{f\in\mathcal{E}(X):g(B_f)=g\big\}.$$

Here $\Pi_g$ denotes the fundamental group of a genus-$g$ curve, and $\theta_q$ denotes the Chen rank. Chen ranks conjecture for Kähler groups. For all $q\gg 0$,

$$\theta_q\bigl(\pi_1(X)\bigr)=\sum_{f\in\mathcal{E}(X)}\theta_q(\pi_1(B_f))=\sum_{g\geq 2}h_g(X)\theta_q(\Pi_g)$$

and hence

$$\theta_q\bigl(\pi_1(X)\bigr)=(q-1)\sum_{g\geq 2}h_g(X)\binom{2g+q-2}{q}-\sum_{g\geq 2}h_g(X)\binom{2g+q-3}{q-2}.$$

This is proposed as the natural Chen-ranks formula for Kähler groups, conditional in the surrounding discussion on the expected reducedness of their projective resonance.