Having same issue here - with same black-hole responsiveness - FOUR YEARS LATER. Obviously Customer Care is not much of a priority at Dell. Given their dated, piecemeal hard-to-navigate and fractious website, this seems to be part of their corporate identity and business model and thus less of a surprise and more of a sad state of affairs for such decent products. and no recourse except for India-located phone banks - which are part of the initial problem and ergo not part of the necessary solution.
This last observation comes as a result of placing a large (for us) home order but only being able to find anywhere on the site items which have shipped already and therefor have a service code. The remainder must be taken on trust as they do not show up and there seems to be no mechanism for ascertaining status and prospects other than the original purchase hard copy info.