Magnetic skyrmions are promising candidates as elementary nanoscale bits in logic-in-memory devices, intrinsically merging high-density memory and computing capabilities. Here we exploit the dynamics of skyrmions interacting with anisotropy energy barriers patterned by ion irradiation to design programmable logic gates. Using micromagnetic simulations with experimental parameters, we show that a fine tuning of the barrier height and width allows the selective tunneling of skyrmions between parallel nanotracks triggered by skyrmion-skyrmion interaction. This can be leveraged to design a skyrmion demultiplexer logic gate that works solely using skyrmions as logic inputs. By cascading and connecting demultiplexer gates with a specific topology, we develop a fully programmable logic gate capable of producing any possible logic output as a sum of all minterms generated by a given set of inputs without requiring any complex additional electric or magnetic interconversion. The proposed design is fully conservative and cascadable, enabling purely skyrmion-based logic-in-memory devices.