The two critical temperatures conundrum in La$_{1.75}$Sr$_{0.125}$CuO$_4$

The in-plane and out-of-plane components of the stiffness tensor in LSCO, show different transition temperatures, with strong variations of the interplane stiffness on sample width. Disorder and critical finite size corrections are too small to explain these effects. With evidence from Monte Carlo simulations, we show that due to the high anisotropy, a three dimensional sample approaching the transition temperature $T_c$ acts as a quasi one dimensional Josephson array. As such, the interplane stiffness exhibits an essential singularity $\sim exp(-A/|T-T_c|^{2\beta})$. At finite experimental or numerical resolution, the interplane stiffness always {\it appears} to vanish at a lower temperature than the in-plane stiffness. An analogy to studies of helium superfluids in nanopores is made.