A fake is an object which has been made with the intention to deceive or to defraud. Sometimes it is difficult to prove such intention. However, in this case, the fake signature cane is clear evidence of this intent to deceive.
Actually, I am really annoyed at Boldon auctions galleries as I had sent them a message to point out that it was a fake Paul Ysart weight. Obviously, this was completely ignored. Next time, I will make sure to discuss this on the GMB so no member gets caught.
I must admit I am very surprised at the auction house's attitude. If it had been a fake painting for instance, I suspect that it would have been a different story.
Still in your position, I would be tempted to get back to them and see what they say. In my (very) humble opinion, they come close to fall foul of the Fraud Act 2006.