Is there a Limit to the Multiplicative Persistence of a Number?

Theorem
It has been conjectured that there may be an upper limit to the multiplicative persistence of a natural number.

Progress
It is not known whether there exists a number $n$ such that:
 * $\map P n = 12$

where $\map P n$ denotes the multiplicative persistence of $n$, but it is known that it is greater than $10^{200}$.