One Perfect Number for Each Number of Digits

Conjecture
There is one perfect number for each number of digits:
 * $1$ digit: $6$
 * $2$ digits: $28$
 * $3$ digits: $496$
 * $4$ digits: $8128$

Refutation
The $5$th perfect number is $33 \, 550 \, 336$, which clearly does not have $5$ digits.

Also see

 * Last Digit of Perfect Numbers Alternates between $6$ and $8$