  Classically, the inner horizon of a perturbed, rotating black hole undergoes
an instability known as mass inflation, wherein the spacetime curvature
diverges as a result of hyper-relativistic crossing streams of ingoing and
outgoing radiation. The generic outcome of this instability is currently
believed to be a strong, spacelike singularity, potentially alongside a weak,
null singularity surviving at late times. However, the quantum back-reaction in
this regime has yet to be fully calculated for a realistic black hole
spacetime. Here we consider a massless quantized scalar field $\phi$ over the
inflationary Kasner spacetime, a recently developed model for the inner horizon
geometry of a rotating, accreting black hole. With this spacetime, we use
numerical adiabatic regularization to calculate
$\langle\phi^2\rangle_\text{ren}$, the renormalized coincidence limit of the
two-point correlation function, as a pointer to the behavior of the quantum
stress-energy tensor. $\langle\phi^2\rangle_\text{ren}$ is generically found to
be nonzero near the inner horizon, divergent where the curvature classically
diverges, and larger for smaller black hole spins or accretion rates.
