Open, for the three special questions (iii.b)--(iii.d), on which nothing was found. Three of the six questions are answered no: (i) and (ii) by the site's construction (a sequence of primes $a_k$ with $a_k+k\equiv0\pmod{q_k}$, or $\pmod{q_k^2}$, for primes $q_k\nmid k$, which grows as fast as desired and leaves $0$ as the only integer shift making every term prime, and at most $0$ and $1$ among the nonnegative shifts for squarefree; the construction has been on the page since its edit of 8 April 2026 and is the curator's own pending partial claim, the site's construction; a forum note of 15 April 2026 gives a version with exactly one integer shift and a Lean formalization); and (iii.a) by an argument on the multiplicative order of $2$, which the site credits to the note's author and GPT (GPT Pro, in the author's own thread comment), which was already the official solution of Problem 4 of the 2015 ELMO competition for every shift $n\ge2$ (its claim page (Gurev Korsky, 2015)), and which the same note proves for all $n$ (even $n$; $n=1$ by $641\mid2^{32}+1$; odd $n\ge3$ by the order argument), with a Lean file, cited by the formal-conjectures collection, that is not built here. So the label attaches to the remaining special questions while the two general questions and the always-prime question are answered; the three answers are recorded on the page, on the pending partial claim page Barschkis's negative answers and on the curator's pending partial claim page, and the standing in the frontmatter, derived from full claims only, stays open. No refereed source exists for any of the answers; they rest on the site's commentary (by the site's revision history, the construction since the edit of 8 April 2026 and the order argument since the edit of 17 April 2026, made in response to the note), the forum note and the elementary checks recorded below.