No, because of 1…axb1.
Or, for that matter, 1…Kxb1.
Bill Smythe
I actually have this problem set up now on two boards at my house, and have offered my kids $50 if they can solve it (all permutations) with no computer or reference resources.
I have not succeeded in interesting them in chess very much to date, but they know how to play - and this intrigues them (the money more than the puzzle… for now).
It is certainly not the hardest puzzle I’ve seen, but likely the most interesting and elegant. They are unlikely to get it, but I will be happy to pay if one of them actually does!
Bill, I finally remembered to bring this problem to my chess club last night. What a hoot! I know you’d have been amused to see how much interest it generated. I had intended simply to had out copies, offering tips/solutions via email, later, if people were interested. Instead, one by one, they put their pre-game discussions and skittles aside, and starting working on the problem. When we were going to start our serious games a little later, it was hard to get their attention to do so! Two experts and a master, plus about 15 other club members, were all debating the problem. In every case, they assumed a Queen was involved.
Eventually one guy who is about my own playing strength, standing behind the others at the board, blurted out: “maybe you have to use a Knight!”, but they scoffed at him and ignored the idea. Too funny! And then after the games, more of the same - well past 11pm. A few asked me to show them the solution; for the rest, I promised to unveil it next week.
Thanks again for posting it here, several weeks ago now. It’s generated a lot of fun.
Of course, now you’ve got to assume all your kids are very much on the honor system and that none of them are (or become…) forum members. 
And it is a very elegant puzzle. Must remember to take it along with me on Saturday. 
Hah! I’d like to indeed assume that they’re on ‘honor system’, yes; but it’s a moot point - as any possible honor (or dishonor) has been trumped, by their preferred ‘lazy system’! 
In other words I would have paid to see the initiative, even if it were a bit shady. They were sworn not to use a computer to solve it (directly), but could have enlisted their friends, as in one case my son did. If they looked on the forum I think I would have judged that that was OK, too (obviously ‘winning’), but I guess they didn’t think to try to figure out where I found the problem in the first place. 
I gave them a little more than a week, which of course has run out. 
The money is now ‘off the table’. In the end, two of the three at home found the solution mildly interesting, at least. So there’s a ray of hope! 
Hey, can you tell I like these ‘smilies’? 
But alas I didn’t get to use my favorite one in this note.
Oh WAIT, it’s about to pop out… D’OH… 
Ahhhh… better now.
I wasn’t at No Exit that night, but I remember your showing the problem to several of us 30(?) years ago at Jules Stein’s.
Try an easier example. Try the last example (page 22 of December 2009 CL4K).
main.uschess.org/content/view/9899/469/
By last example I think you mean the, "Here is an ingenious little composition for you to ponder. " … ?
And do you want answers to that puzzle posted to this thread, or does it not matter? (Don’t want to spoiler it out if you really expected it to be pristine for CL4K readers… Then again, my interpretation of it could be wrong. )
My Gnomes from Outer Mongolia won’t mind answering a few more emails.
In the final CL4K position, the pawns on d5 and e4 make it too obvious. The Smytheian position is (for once!) more natural than the monstrosity on the b file.
But I do like the frustration of the Nf7: it ineffectually stops mate on b7 (stops one kind, allows another) & ineffectually covers h8.
And the idea is cute.
Tim Krabbé’s underpromotions page was cited upthread.
He’s also got a Babson Task page right here.