Femtoscopic correlations with the particle pair combinations KS0KS0 and KS0K± are studied in pp collisions at s=5.02 and 13 TeV by the ALICE experiment. At both energies, boson source parameters are extracted for both pair combinations, by fitting models based on Gaussian size distributions of the sources, to the measured two-particle correlation functions. The interaction model used for the KS0KS0 analysis includes quantum statistics and strong final-state interactions through the f0(980) and a0(980) resonances. The model used for the KS0K± analysis includes only the final-state interaction through the a0 resonance. Source parameters extracted in the present work are compared with published values from pp collisions at s=7 TeV and the different pair combinations are found to be consistent. From the observation that the strength of the KS0KS0 correlations is significantly greater than the strength of the KS0K± correlations, the new results are compatible with the a0 resonance being a tetraquark state of the form (q1,q2‾,s,s‾), where q1 and q2 are u or d quarks.