As Alan has said, the canes in the stopper are definitely Whitefriars (pre-Caithness). John's comment about a sandblasted mark is unlikely to refer to a "cane", but the Museum of London book, "Whitefriars Glass" seems to make no mention of what type of mark it could be.
No question about the ID, unless by chance somebody has put a Whitefriars stopper with another company's unadorned bottle.