Stellar-mass black holes descend from high-mass stars, most of which had stellar binary companions. However, the number of those binary systems that survive the binary evolution and black hole formation is uncertain by multiple orders of magnitude. The survival rate is particularly uncertain for massive stars with low-mass companions, which are thought to be the progenitors of most black hole X-ray binaries. We present a search for close black hole companions (orbital period <~3 days, equivalent to separation <~20R_{sun}_) to AFGK-type stars in TESS, i.e. the non-accreting counterparts to and progenitors of low-mass X-ray binaries. Such black holes can be detected by the tidally induced ellipsoidal deformation of the visible star, and the ensuing photometric light-curve variations. From an initial sample of 4.7x10^6^ TESS stars, we have selected 457 candidates for such variations. However, spectroscopic followup of 251 of them shows that none are consistent with a close black hole companion. On the basis of this non-detection, we determine (2{sigma} confidence) that fewer than one in 10^5^ Solar-type stars in the Solar neighbourhood host a short-period black hole companion. This upper limit is in tension with a number of "optimistic" population models in the literature that predict short-period black hole companions around one in ~10^4-5^ stars. Our limits are still consistent with other models that predict only a few in ~10^7-8^.