  We consider wetting models in $1+1$ dimensions on a shrinking strip with a
general pinning function. We show that under diffusive scaling, the interface
converges in law to to the reflected Brownian motion, whenever the strip size
is $o(N^{-1/2})$ and the pinning function is close enough to critical value of
the so-called $\delta$-pinning model of Deuschel, Giacomin, and Zambotti
[DGZ05]. As a corollary, the same result holds for the constant pinning strip
wetting model at criticality with $o(N^{-1/2})$ strip size.
