We present measurements of the thermal conductivity λ(t,P,L)=l/ρ(t,P,L) near the superfluid transition of 4 He at saturated vapor pressure and confined in cylindrical geometries with radii L=0.5 and 1.0 μm (t≡T/T λ (P)−1). For L=1.0 μm measurements at six pressures P are presented. At and above T λ the data are consistent with a universal scaling function F(X)=(L/ξ o ) x/ν (ρ/ρ 0 ),X=(L/ξ o ) 1/ν t valid for all P (ρ 0 and x are the pressure-dependent amplitude and effective exponent of the bulk resistivity ρ(t,P,∞)=ρ 0 t x and ξ=ξ 0 t −ν is the correlation length). Indications of breakdown of scaling and universality are observed below T λ .