  We study the inclusive strange axial-vector meson $K_1(1270)^+$ production in
${K^-}A$ reactions at near-threshold laboratory incident antikaon momenta
within a nuclear spectral function approach, which describes incoherent direct
$K_1(1270)^+$ meson production in $K^-$ meson--proton ${K^-}p \to
{K_1(1270)^+}\Xi^-$ production processes and accounts for three different
options for its in-medium mass shift (or for its effective scalar potential) at
central density $\rho_0$. We calculate the absolute differential and total
cross sections for the production of $K_1(1270)^+$ mesons on $^{12}$C and
$^{184}$W target nuclei at laboratory angles of 0$^{\circ}$--45$^{\circ}$ by
$K^-$ mesons with momenta of 2.5, 2.8 and 3.5 GeV/c, which are close to the
threshold momentum ($\approx$ 2.95 GeV/c) for $K_1(1270)^+$ meson production
off the free target proton at rest. The intrinsic properties of carbon and
tungsten target nuclei have been described in terms of their spectral
functions, which take into account the momenta of target protons and the
energies of their separation from the considered nuclei. We show that the
differential and total (absolute and relative) $K_1(1270)^+$ antikaon-induced
production cross sections at initial momenta not far from threshold -- at
momenta $\sim$ 2.8--3.5 GeV/c, at which there is yet no strong drop in their
strength, reveal a distinct sensitivity to changes in the in-medium shift of
the $K_1(1270)^+$ mass, studied in the paper, both in the $K_1(1270)^+$ meson
low-momentum (0.1--1.0 GeV/c) and in its full-momentum ranges. This would
permit evaluating this shift. Experimental data necessary for this aim can be
obtained in a dedicated experiment at the J-PARC Hadron Experimental Facility.
