Thermal conductivity of thin films usually decreases as their thickness reduces. However theoretically it is possible to take advantage of surface effects such as surface phonon-polaritons to observe an increase of this quantity for certain thin films such as amorphous glass membranes. Surface phonon-polaritons are evanescent electromagnetic waves propagating along the surface of polar dielectric materials and can be very powerful heat carriers for nanofilms due to their long propagation length. This paper aims to prove it experimentally. Free-standing amorphous silica films have been successfully prepared by dry oxidation with thicknesses ranging from 20 to 300 nm. We have used the AC calorimetry method to measure the in-plane thermal conductivity of the samples. Our preliminary results show an increase of the thermal conductivity for thicknesses smaller than 200 nm. This is in agreement with the theory, however this increase is not monotonous and the measured values are lower than theoretical predictions. This is probably due to the film sizes, which are restricting the propagation of surface phonon-polaritons. Comparisons between the width of the membrane and the in-plane thermal conductivity confirm this trend through the thermal conductivity increase with the film width, for a constant film thickness.