Let $r$ be a prime power. The Topological Tverberg Theorem says that any continuous map from a $(r-1)(d+1)$-simplex to $\mathbb{R}^d$ identifies points from $r$ disjoint faces. It is not hard to see that we cannot replace $\mathbb{R}^d$ with $\mathbb{R}^{d+1}$ in Tvereberg's theorem. Regarding this theorem, consider the following definition.

For a simplicial complex $K$ and a positive integer $r\geq 2$, let $T(K, r)$ be the minimum $t$ such that there is a continuous function from $K$ to $\mathbb{R}^t$ which does not identifies points from any $r$ disjoint faces. Therefore, Tvereberg's theorem in this setting says $T(\Delta_{(r-1)(d+1)}, r)= d+1$ whenever $r$ is a prime power.

1) Is there any research paper exists for determining $T(K, r)$ for other simplicial complexes rather than a simplex?

2) Is $T(K, r)=d+1$, if $K$ is homeomorphic with $\Delta_{(r-1)(d+1)}$?

Please note that, for the second question, we have $T(K, r)\geq d+1$ as any such a simplicial complex contains a $(r-1)(d+1)$-simplex.