Asymptotically optimal colorful Tverberg tolerance conjecture

Let $r,d$ be fixed positive integers. A colorful partition of $M$ families of $r$ points each in $\mathds{R}^d$ is a partition into $r$ sets $A_1,\ldots,A_r$ assigning one point from each family to every part. The asymptotically optimal colorful Tverberg tolerance conjecture. There is an integer

$$M=M(t,d,r)=t(1+o(1))$$

such that, given any $M$ families, for any family $C$ of at most $t$ colors,

$$\bigcap_{j=1}^r \operatorname{conv}(A_j\setminus C)\neq\emptyset.$$

The conjecture asks whether the constant in the paper's preceding theorem can be improved to $1$ asymptotically; it is presented as an interesting unresolved question.