VCD vs. RTD

Definitions

Given $n$, let $2^n$ be the set of all $n$-bits binary strings, and $[n]=\{0,1,\cdots,n-1\}$. Given a class $\mathcal{C}\subset 2^n$.

The VC dimension, $\text{VCD}(\mathcal{C})$ is the maximum $|A|$ such that each binary string on $A$ can be extended to a member in $\mathcal{C}$, i.e.

$$\text{VCD}(\mathcal{C})=\max\{|A|:A\subset [n],|\{c\upharpoonright_A:c\in\mathcal{C}\}|=2^{|A|}\}$$

The best-case teaching dimension, $\text{TD}_{min}(\mathcal{C})$ is the minimum $|B|$ such that the restriction of some $c\in\mathcal{C}$ on $B$ is enough to identify that element in $\mathcal{C}$, i.e.

$$\text{TD}_{min}(\mathcal{C})=\min\{|B|:B\subset[n],\exists c\in\mathcal{C},|\{c’\in\mathcal{C}:c’\upharpoonright_B=c\upharpoonright_B\}|=1\}$$

Known facts

Fact. $\text{TD}_{min}(\mathcal{C})=O(\text{VCD}(\mathcal{C})^2)$.

Fact. There is $\mathcal{C}$ such that $\text{TD}_{min}(\mathcal{C})=5n$ and $\text{VCD}(\mathcal{C})=3n$. This is the best known worst-case lower bound.

The problem

Question. Is it true that $\text{TD}_{min}(\mathcal{C})=O(\text{VCD}(\mathcal{C}))$?