What makes you think it's the second generation? I know these two smiths particularly well, and I know what to look for to tell which generation this sword belongs to.
Generally, when a swordsmith takes the trouble to make an umegane, which is not an easy thing to do, it is because the quality of the sword justifies it; otherwise, he destroys it and starts over.
Thank you, but I never give an opinion without having held the sword in question in my hand. I ad I'm not an expert, just what you might call an informed amateur.
Okay,
In fact, Hon'ami Kozon does not say that there is only one tachi, but that he has only seen one. Have you had the opportunity to see both the tachi you mention and a tanto at the same time to judge the potential difference in quality? Do you have the knowledge to evaluate such a thing? Personally, I don't, and I trust a real expert more than an ordinary person who claims to be one.
I questionned Chatgpt about the number of tachi answer below