Cell-complex realization conjecture for spheres with $g_k=1$

Let $k\geq 2$, $d\geq 2k$, and let $\Delta$ be a $(d-1)$-sphere with $g_k(\Delta)=1$. Cell-complex realization conjecture. There exists a $d$-dimensional cell complex $\mathcal{C}$ such that all faces of $\mathcal{C}$ of dimension at most $d-k$ are faces of $\Delta$, all faces except one $d$-face are simplices, and the exceptional $d$-face is a homology $d$-ball whose boundary is a sphere in $S(d-k,d-1)$ with $g_k=1$. Every two faces of $\mathcal{C}$ intersect in a common, possibly empty, face; the geometric realization of $\mathcal{C}$ is a homology $d$-ball; and its boundary complex is $\Delta$. Furthermore, if $\Delta$ is the boundary complex of a simplicial $d$-polytope, then the exceptional $d$-cell is also the boundary complex of a simplicial $d$-polytope.

The conjecture is proposed as a possible characterization of all $(d-1)$-spheres with $g_k=1$, without assumptions on the dimensions of missing faces, and is motivated by the generalized lower bound theorem. Its resolution status is not specified in the source.