Unbounded ratio of maximum throttling number to maximum cop number

For each positive integer $n$, let $mc(n)$ and $mt(n)$ denote, respectively, the maximum cop number and maximum cop throttling number over all connected graphs of order $n$. The unbounded-ratio conjecture.

$$\lim_{n \rightarrow \infty} \frac{mt(n)}{mc(n)}=\infty.$$

The claim predicts that maximum cop throttling can grow arbitrarily faster than maximum cop number. The surrounding discussion introduces this as a further question after establishing bounds on cop throttling, but the supplied text gives no resolution.