On vertex-minimal simplicial maps to the sphere

For positive integers $n,d$, let $\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere which admits a degree $d$ simplicial map onto the boundary of $(n+1)$-simplex.

We show that for $h=\lfloor\frac{n+1}2\rfloor$, the function $\lambda(n,d)^h$ is almost linear in $d$ as $d\to\infty$ answering a question by this http URL.

All triangulations we obtain are isomorphic to boundaries of convex polytopes in $\mathbb{R}^{n+1}$.