The sparse simplicial-complex index conjecture

Let $\mathcal H\subset 2^{[n]}$ be a simplicial complex, and let $\mathcal H^{*2}_{\Delta}$ be its deleted join. Its $\mathbb Z_2$-index, denoted $\mathrm{ind}_{\mathbb Z_2}(\mathcal H^{*2}_{\Delta})$, is the least $d$ for which there is a $\mathbb Z_2$-equivariant map from its polyhedron to $S^d$. Sparse simplicial-complex index conjecture. There exists a constant $c>1$ such that, whenever $\mathcal H$ has fewer than $c^k$ $k$-element sets,

$$\mathrm{ind}_{\mathbb Z_2}(\mathcal H^{*2}_{\Delta})\leqslant 2k-3.$$

Moreover, the stronger assertion that $\mathcal H$ is embeddable into $\mathbb R^{2k-3}$ may hold. The conjecture is presented as implying the preceding exponential-deficit conjecture; the supplied text gives no resolution.