A connected special étale cover stays connected on the geometric generic fibre

If $Y_k$ is empty, properness forces $Y$ to be empty: a nonempty closed image in the local trait would contain its closed point. Its geometric generic fibre is then empty and hence connected under the library convention. Assume now that $Y_k$ is nonempty. For any finite separable extension $L\mathrm{/}K$, its integral closure $R^{\mathrm{\prime }}$ over $R$ is finite by [F2]. It is complete and semilocal, with $R^{\mathrm{\prime }}\mathrm{/}t{R}^{\mathrm{\prime }}$ Artinian. If it had two local factors, their nontrivial idempotent would lift by henselianity in [F2], contradicting the fact that $R^{\mathrm{\prime }}$ is a domain. Thus it is local, normal and one-dimensional, hence a complete DVR by [F2]. Its residue field is a finite extension of algebraically closed $k$, so is $k$. The special fibre of $Y_{R^{\mathrm{\prime }}}$ is therefore $Y_k$ and is connected. Any open and closed decomposition of proper $Y_{R^{\mathrm{\prime }}}$ would give two nonempty closed images in the local trait, each meeting its closed point, contradicting that connected special fibre. Hence the total space is connected. It is regular by [F1], thus integral by [F3], and its generic fibre $Y_L$ is connected.

If $Y_{\bar{K}}$ were disconnected, its open and closed decomposition would be witnessed by a nontrivial idempotent in its global function sheaf. This finite datum descends to a finite field extension: take a finite affine cover of $Y_K$ and finite affine covers of its intersections, write the idempotent as local ring elements with their equalities, and put the finitely many coefficients occurring in those elements and relations in a finite extension $L\mathrm{/}K$. The equations $e^2=e$, agreement on overlaps and nontriviality then hold after a sufficiently large finite extension; nontriviality is detected by faithful field base change. The algebraic closure is the union of its finite extensions. Purely inseparable finite extensions preserve the underlying topology: their affine tensor spectra have unique radicial points over each residue-field point, and the extension is integral and surjective. Thus disconnectedness already occurs after the separable part of a finite extension, contradicting step 1.1. Therefore $Y_{\bar{K}}$ is connected.

By [F1], every connected finite étale cover of $X$ has connected closed fibre, since a disconnected closed cover would lift its open and closed components to a decomposition upstairs. Step 2.1 therefore preserves every connected cover on the geometric generic fibre. By [F3] this makes the induced homomorphism of profinite groups surjective: otherwise its compact image $H$ is a proper closed subgroup, and some finite quotient $G\mathrm{/}N$ has $HN\mathrm{\ne }G$. The corresponding connected cover with regular transitive fibre $G\mathrm{/}N$ becomes disconnected when restricted to $H$, contradicting the just-proved connectedness. Finally [F1]'s cover equivalence identifies ${\pi }_1(X)$ with ${\pi }_1(X_k)$ after selecting a path between the two fibre functors. Such a path exists by taking compatible points of the second fibre functor on the pointed Galois system in [F3]; its transition maps are surjective and its finite inverse limit is nonempty by the compactness step there. Different basepoint paths conjugate the target identification. This yields the asserted specialization surjection.