Map folding ($3 \times 3)$

The number of distinct ways to fold a $3 \times 3$ rectangle into a unit square.

Updates

• Lower bound: $1368$
  Reference unknown
  [via Number of ways of folding an n X n sheet of stamps. - OEIS]
• Upper bound: $1368$
  Reference unknown
  [via Number of ways of folding an n X n sheet of stamps. - OEIS]