How long can the primitive-set saturation game be forced to last? (Erdős #872)

Two players alternately pick integers from $\{2,3,\ldots,n\}$, each pick added to a common set $A$, subject to the rule that $A$ must stay primitive: no chosen element may divide another (no $a\mid b$ for distinct $a,b\in A$). Play ends when no legal move remains — i.e. $A$ is a maximal primitive subset of $\{2,\ldots,n\}$. One player (the 'Prolonger') tries to make the game last as long as possible; the other (the 'Shortener') tries to end it quickly. Under optimal play by both, how many moves can the length be guaranteed to be? In particular, is the guaranteed length linear — at least $\epsilon n$ for some fixed $\epsilon>0$ and all large $n$ — and can one even guarantee $(1-\epsilon)\tfrac{n}{2}$ moves?

Acceptance. FULLY RESOLVES: settle whether the guaranteed game length is linear in $n$ under a specified first-move convention — either a full proof (with an explicit Prolonger strategy) that it is at least $\epsilon n$ for some fixed $\epsilon>0$ and all large $n$, or a full proof (with an explicit Shortener strategy) that it is $o(n)$. ADVANCES: improve either side of the current window with proof, anchored to the bounds stated in background — a proved lower bound strictly above $\gg n/\log n$ (e.g. $\geq n(\log n)^{1+c}$ or linear), or a proved Shortener upper bound strictly below the $(\tfrac{23}{48}+o(1))n$ stated in background; OR compute, by exhaustive minimax, the exact guaranteed length for a new range of small $n$ (delivering the game-solver code and the table of values, stating the first-move convention) as data toward the linearity question. Always state which player moves first. Deliver the strategy plus proof, or the improved constant plus proof, or the game-solver plus certified exact values.