The asymptotic simplicial mapping conjecture for spheres

Let $T_1$ be a triangulation of the $n$-sphere and let $S_{n+2}^n$ be the boundary triangulation of the $(n+1)$-simplex. For a coloring $L:V(T_1)\to V(S_{n+2}^n)$, let $f_L:\mathbb S^n\to\mathbb S_{n+2}^n$ be the induced simplicial map. Define $\alpha(n):=\limsup_{d\to\infty}\lambda(n,d)/d$, where $\lambda(n,d)$ is the smallest number of vertices of a triangulation admitting such a coloring with $\deg f_L=d$. Asymptotic simplicial mapping conjecture.

$$\alpha(n)=\frac{n+2}{n}.$$

The paper proves the upper bound $\alpha(n)\leq (n+2)/n$; the conjecture asserts that this bound is sharp for every $n$.