Map folding ($2 \times 4)$

The number of distinct ways to fold a $2 \times 4$ rectangle into a unit square.

Updates

• Lower bound: $320$
  Reference unknown
  [via Number of ways of folding a 2 X n strip of stamps. - OEIS]
• Upper bound: $320$
  Reference unknown
  [via Number of ways of folding a 2 X n strip of stamps. - OEIS]