Unlike the widely studied $s$-type two-gap superconductor MgB$_2$, the chemically similar compounds ZrB$_2$ and HfB$_2$ do not superconduct above 1 K. Yet, it has been shown that small amounts of self- or extrinsic doping (in particular with vanadium), can induce superconductivity in these materials. Based on results of different macro- and microscopic measurements, including magnetometry, nuclear magnetic resonance (NMR), resistivity, and muon-spin rotation ($mu$SR), we present a comparative study of Zr$_{0.96}$V$_{0.04}$B$_2$ and Hf$_{0.97}$V$_{0.03}$B$_2$. Their key magnetic and superconducting features are determined and the results are considered within the theoretical framework of multiband superconductivity proposed for MgB$_2$. Detailed Fermi surface (FS) and electronic structure calculations reveal the difference between MgB$_2$ and transition-metal diborides.
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