Open: the site's label (page last edited 27 December 2025). The site's commentary credits Kevin Barreto's thread post of 16 December 2025 with a conditional answer to the second question. The post assumes two conjectures. The first strengthens the Lagarias--Soundararajan $xyz_{\mathrm{fin}}$ conjecture: for some $\kappa_0>1$ and every $\epsilon>0$, only finitely many coprime $X+Y=Z$ have every prime factor of $XYZ$ below $(\log\max(|X|,|Y|,|Z|))^{\kappa_0-\epsilon}$. The second is a lower bound $\log n\ge ck^{\alpha}$, with $\alpha>1/\kappa_0$ and all large $k$, on the least $n\ge2k$ at which $\binom nk$ has deficiency at least $2$, when such an $n$ exists. From these the post proves that only finitely many $\binom nk$ with $n\ge2k$ have deficiency at least $2$. A conditional thread post, it has no claim page.