2004 Honda Accord starting issue

My 2004 Honda Accord is having starting issues.  When i try to start my car
in the morning (garage kept) is takes a very long time to start.  Once the car
finally starts it drives normal.  The problem begins to increase during the rest
of the day.  When I try to restart the car it takes 3 times to start.  The 1st and
2nd times are bad.  the car just cranks with NO start. The 3rd time that I turn
the key the car starts normal.  I recently took the car to the dealership for a
diagnostic.  The dealership was unsure of the problem.  They guess it
maybe the fuel pump going bad.  I am not sure I trust them.  I was told my
the dealer mechanic that there's no guarantee that a fuel pump replacement
will resolve my issue.