  To identify progenitors and investigate evidence of He burning, we searched
for decay radiation of freshly synthesized $^{44}$Ti in four young nearby
thermonuclear supernova remnants: Kepler, SN 1885, G1.9+0.3 and SN 1006, by
analysing the up-to-date NuSTAR archival data. No apparent flux excess from the
68 and 78 keV line emissions accompanying decay was detected above the power
law continuum applied for the remnants and the absorbed stray light. By
comparing the inferred upper limits of the line flux and the initial $^{44}$Ti
masses with a wide variety of supernova nucleosynthesis models, we placed
constraints on the supernova progenitors. We derived the first NuSTAR line flux
upper limit for Kepler and ruled out most of the double-detonation scenarios
with a thick He layer under low density. We estimated, for the first time, the
upper limit for SN 1885, which is high because of the large distance yet still
remains consistent with the He shell detonation. The new flux and mass limit of
G1.9+0.3 derived from a longer total exposure is lower than the results from
previous studies and evidently excludes explosive burning of He-rich matter.
The relatively advanced age and the large spatial extent of SN 1006 have
prevented meaningful constraints.
