Akita's integral Riemann–Roch conjecture for surface bundles

Let $\pi\colon E\to B$ be an oriented surface bundle with closed fibers of genus $g$, equipped with a fiberwise metric. Let $\mathcal{H}(E)$ be the associated Hodge bundle, with fibers $H^1(E_b;\mathbb{R})$, regarded as a $g$-dimensional complex vector bundle via the complex structure induced by the Hodge star operator. Define

$$s_n(E):=n!\operatorname{ch}_n\bigl(\mathcal{H}(E)\bigr)\in H^{2n}(B;\mathbb{Z})$$

and

$$\kappa_n(E):=\pi_*\bigl(c_1(T^\pi E)^{n+1}\bigr)\in H^{2n}(B;\mathbb{Z}).$$

Here $\operatorname{Num}(r)$ and $\operatorname{Denom}(r)$ denote the numerator and denominator of a rational number $r$ in lowest terms. Akita's integral Riemann–Roch conjecture. In $H^*(B;\mathbb{Z})$,

$$\operatorname{Denom}(B_n/2n)\,s_{2n-1}(E)=(-1)^{n-1}\operatorname{Num}(B_n/2n)\,\kappa_{2n-1}(E).$$

This conjecture asks whether the rational family Riemann–Roch relation can be cleared of denominators integrally for every oriented surface bundle. The paper is presented as a response to Akita's conjecture; the supplied text does not establish whether the conjecture is resolved.