Numbers for which Euler Phi Function equals Product of Digits

Theorem
The sequence of positive integers $n$ for which $\map \phi n$ is equal to the product of the digits of $n$ begins:
 * $1, 24, 26, 87, 168, 388, 594, 666, 1998, 2688, 5698, 5978, 6786, 7888, 68 \, 796$

It is known that this sequence is finite, but it is unknown whether a $16^{\text {th}}$ term exists.

Proof
and so on.