Immersion and embedding bounds for manifolds over properly colored polytopes

Let $P^n$ be a simple convex $n$-dimensional polytope that admits a proper coloring with $n$ colors. Let $\alpha(n)$ denote the number of $1$'s in the binary expansion of $n$, and let $M^n$ be a small cover and $M^{2n}$ a quasitoric manifold over $P^n$. Immersion and embedding conjecture. There exist such manifolds satisfying

$$imm(M^{n})=2n-\alpha(n),\qquad em(M^{n})=2n-\alpha(n)+1,$$ $$imm(M^{2n})\geq 4n-2\alpha(n),\qquad em(M^{2n})\geq 4n-2\alpha(n)+1.$$

This extends the examples constructed in the paper from particular properly colored polytopes to every simple $n$-dimensional properly $n$-colored polytope. The conjecture concerns sharp immersion and embedding behavior for small covers and quasitoric manifolds, while the supplied text gives no evidence that it has been resolved.