I found the reason : in PHP '01' equals '1'
(yes, even if field is a string, numeric values are compared)
So trying to update '1' to '01' gives no result.
You can try manually : it is not linked to the import function, nor to the field.
For instance create an item (any, customer, ticket project); with name = '1'.
Try and change name to '01' => no change

To have a change, you must change you must chamge name to '2' (or anything else different from 1) then to '01'.
It is strange nobody reported this issue before.
But
There is no quick fix for this.
Ticket #1788 recorded to think about a fix.