The Hasse principle for diagonal forms restricted to a hypersurface of adjacent degree

We investigate the Hasse principle for Diophantine systems consisting of one diagonal form of degree $k$ and one general form of degree $k-1$. By refining the method of Brandes and Parsell (arXiv:2003.04350) in this specific setting, we improve the bound $n>2^{k}k$ to $n>2^{k-1}(2k-1)$; in particular, the requirement $n>24$ in the case of degrees three and two is relaxed to $n>20$.