Kontsevich formula conjecture for thin-monodromy variations of Hodge structures

Let $\mathcal{H}^n$ be a weight $n$ variation of Hodge structures with thin monodromy over a hyperbolic Riemann surface $C$ with singular locus $S$. Let $F^{\lceil \frac{n}{2}\rceil}$ denote the corresponding Hodge filtration subbundle, and let $\mathcal{H}^{n,0}$ denote the $(n,0)$-part of the variation. Suppose $\lambda_1,\ldots,\lambda_k$ are the positive Lyapunov exponents. Kontsevich formula conjecture. If $k=\dim F^{\lceil \frac{n}{2}\rceil}$, then

$$\lambda_1+\ldots+\lambda_k=\frac{2\int_C c_1(F^{\lceil \frac{n}{2}\rceil})}{2g(C)-2+\#S}.$$

If $k<\dim F^{\lceil \frac{n}{2}\rceil}$ but $k>\dim \mathcal{H}^{n,0}$, set $d:=\dim \mathcal{H}^{n,0}$. Then

$$\lambda_1+\ldots+\lambda_d=\frac{2\int_C c_1(\mathcal{H}^{n,0})}{2g(C)-2+\#S}.$$

The conjecture is motivated by the weight-one Kontsevich formula and by examples in which thin monodromy is associated with its validity, whereas arithmetic monodromy can lead to failure. Whether thin monodromy implies the formula in general remains unknown, and the arithmeticity of monodromy groups is itself unresolved in many cases.