dcm2niix handles converting DICOM to NIfTI, so MINC is out of scope. You may want to contact the MINC team. Vladimir Fonov provided a talk on the MINC format, so he may be a good contact
  https://nipy.discourse.group/t/data-format-pro-con-meeting-monday-14-march-15-00-edt/122

For Siemens VA-VE DICOMs, the slice timing is in the proprietary CSA header
   https://nipy.org/nibabel/dicom/siemens_csa.html
Specifically, you will want to extract the `MosaicRefAcqTimes` tag.


Your data is from ~2004 (VA25). The oldest archival dataset I have is from 2007 (VB12). Therefore, I am not sure if VA25 includes the CSA header  
   https://www.nitrc.org/plugins/mwiki/index.php/dcm2nii:MainPage#Archival_MRI