The K-theory presentation conjecture for torus manifold bundles

Let $E\longrightarrow B$ be a principal $T$-bundle, let $X$ be a torus manifold with orbit space $Q$, and let $E(X):=E\times_T X$ be the associated bundle over a base $B$ having the homotopy type of a finite CW complex. Write $m=\chi(X)$, let $\mathfrak{S}:=K^*(B)[x_F:F$ a face of $Q]$, and let $\mathfrak{J}$ be the ideal generated by the relations

$$x_Gx_H-x_{G\vee H}\sum_{E\in G\cap H}x_E$$

and

$$\prod_{i:\langle u,v_i\rangle>0}(1-x_{Q_i})^{\langle u,v_i\rangle}-[\xi_u]\prod_{i:\langle u,v_i\rangle<0}(1-x_{Q_i})^{-\langle u,v_i\rangle},\qquad u\in\operatorname{Hom}(T,S^1).$$

K-theory presentation conjecture. The ring $K^*(E(X))$ is a free $K^*(B)$-module of rank $m=\chi(X)$ and is isomorphic to $\mathfrak{S}/\mathfrak{J}$.

When $B=\operatorname{pt}$, this would give a presentation of the K-ring of $X$ generalizing Sankaran's result; for arbitrary $B$, it would generalize the theorem proved earlier in the paper. The conjecture concerns the K-theory of associated torus manifold bundles and remains unresolved here.