Viehweg–Zuo's factorization and stability conjecture for Arakelov-equality variations

Let $Y$ be a smooth projective variety with boundary divisor $S$ satisfying the assumptions denoted by, and let

$$\Omega_Y^1(\log S)=\Omega_1\oplus\cdots\oplus\Omega_s$$

be the decomposition into stable sheaves. Let ${\mathbb V}$ be an irreducible subvariation of Hodge structures with Higgs bundle

$$\left(E^{1,0}\oplus E^{0,1},\ \theta:E^{1,0}\to E^{0,1}\otimes\Omega_Y^1(\log S)\right),$$

and suppose it satisfies the Arakelov equality

$$\mu_{\omega_Y(S)}(E^{1,0})-\mu_{\omega_Y(S)}(E^{0,1})=\mu_{\omega_Y(S)}(\Omega_Y^1(\log S)).$$

Factorization and stability conjecture. There exists a unique $i\in\{1,\ldots,s\}$ such that the Higgs field factors through

$$\theta:E^{1,0}\to E^{0,1}\otimes\Omega_i\to E^{0,1}\otimes\Omega_Y^1(\log S),$$

and $E^{1,0}$ and $E^{0,1}$ are $\mu_{\omega_Y(S)}$-stable.

The Arakelov equality is known to imply semistability of the two Hodge bundles, but the stronger assertion of stability and factorization through a unique stable summand of the logarithmic cotangent bundle is presented as an expectation for higher-dimensional bases.