The branched-cover conjecture for Garland posets

Let $k>0$. A finite cubical poset $Q$ has dimension $k+1$, and every $k$-face of $Q$ is contained in at least $d$ facets. A $k$-branched cover is a branched cover with branching in the sense used for Garland posets.

Branched-cover conjecture. For every $k>0$ there is $d$ such that, whenever $Q$ is a finite cubical poset of dimension $k+1$ and every $k$-face is contained in at least $d$ facets, there is a pair of $k$-branched covers

$$Q \leftarrow P \rightarrow R$$

such that the Garland poset of $R$ satisfies the hypotheses of the cubical Garland result.

The conjecture proposes a way to overcome the fact that passing to covers can enlarge the spectrum of link graphs and thereby invalidate the required spectral hypothesis. The preceding discussion explains that suitable branched covers can be made $k$-monodromy-free; what remains conjectural is the existence of the pair of covers with the stated Garland property.