Box Progression Part 3

$\large \square + \square\square+\square\square\square+\square\square\square\square$

You are given that the numbers $0,1,2,\ldots,9$ are to be filled in the square boxes as shown above (without repetition) such that it represent a sum of a 1-digit, 2-digit, 3-digit, and 4-digit number.

Find total number of possible arrangements of these nine numbers such that the sum of these four numbers is minimized.

Details and Assumptions:

• This is an arithmetic puzzle, where $1 \square$ would represent the 2-digit number 19 if $\square = 9$. It does not represent the algebraic expression $1 \times \square$.
• The numbers are not allowed to have a leading 0 (even the 1-digit number).
• See Part 1, Part 2 and Part 4.