The NuDoubt⁺⁺ experiment aims at the measurement of two-neutrino and neutrinoless positive double weak decays (2β⁺/ECβ⁺). It is based on a new detector concept combining hybrid and opaque scintillators paired with a novel light read-out technique. The technology is particularly suitable detecting positrons (β⁺) signatures. In its first phase, NuDoubt⁺⁺ is going to operate under high-pressure loading of enriched Kr-78 gas. It expects to discover two-neutrino positive double weak decay modes of Kr-78 within 1 tonne-week exposure and is able to probe neutrinoless positive double weak decay modes at several orders of magnitude improved significance compared to current experimental limits. Later phases may involve searches for positive double weak decays in Xe-124 and Cd-106.