Puzzle 3: Münchhausen numbers of length 3

The number 3435 = 3³+4⁴+3³+5⁵ is well-known as the Münchhausen number in the base 10 and its length (number of digits) is 4.

More examples:

The number 96446 is the Münchhasuen number in the base 9 and its length is 6, because it has 6 digits in the base 9:

96446₁₀  = 156262₉  

digits 1, 5, 6, 2, 6, 2    

96446 = 1¹+5⁵+6⁶+2²+6⁶+2²

The number 20017650854 is the Münchhasuen number in the base 12 and its length is 10, because it has 10 digits in the base 12:

20017650854₁₀  = 3a67a54832₁₂ 

digits 3, a, 6, 7, a, 5, 4, 8, 3, 2  

20017650854 = 3³+10¹⁰+6⁶+7⁷+10¹⁰+5⁵+4⁴+8⁸+3³+2²   (the digit a is ten)

Bases can be considered arbitrary. Then it is not difficult to prove that there are infinitely many Münchhasuen numbers of length 2 in some base. In particular, 1+nⁿ is of length 2 in the base 1+nⁿ-n for each n greater than 1.

But what about length 3? Are there infinitely many Münchhausen numbers of length 3? Up to now, we know only these Münchhasuen numbers of length 3:

29 in base 4

55 in base 4

3153 in base 25

49782 in base 91

46661 in base 215
New: 823545 in base 904 (June 21, 2020; by mersenneforum.org user axn)