  In this work we present new results on the convergence of diagonal sequences
of certain mixed type Hermite-Pad\'e approximants of a Nikishin system. The
study is motivated by a mixed Hermite-Pad\'e approximation scheme used in the
construction of solutions of a Degasperis-Procesi peakon problem and germane to
the analysis of the inverse spectral problem for the discrete cubic string.
