It has previously been suggested an X-ray line of photon-energy 3.55 keV (in the emission frame), not being understood in terms of expected materials in the astrophysical sources, might come from dark matter. We confront this idea within our own model for dark matter. Our model is remarkable by not needing new physics, but only the Standard Model supplemented by a fine tuning law, the "Multiple Point Principle", restricting the values of the coupling constants and the Higgs mass in the Standard Model. The crux of the matter is that in our model the dark matter consists of cm-sized pearls, which are bubbles of a suggested new type of vacuum (called "condensate vacuum") containing some ordinary atomic matter under a very high pressure caused by the weak-interaction-scale surface tension. The inside ordinary matter is hoped to be a glassy insulator with a homolumo gap that happens to be close to the 3.55 keV, so that excitons may decay giving such X-ray radiation. The energy needed for producing this 3.55 keV-radiation is suggested to come from the surface contraction made possible, when two such pearls - very seldomly - hit each other. A major success is that we get the intensity rather well predicted order of magnitudewise in our model, using values of the parameters of our model already set up to fit properties considered before we looked at the 3.55 keV radiation. In addition we make some theoretical considerations on the homolumo-gap model developed by two of us and I. Andric and L. Jonke.