We study semi-parametric estimation of the population mean when data is observed missing at random (MAR) in the $n < p$ "inconsistency regime", in which neither the outcome model nor the propensity/missingness model can be estimated consistently. Consider a high-dimensional linear-GLM specification in which the number of confounders is proportional to the sample size. In the case $n > p$, past work has developed theory for the classical AIPW estimator in this model and established its variance inflation and asymptotic normality when the outcome model is fit by ordinary least squares. Ordinary least squares is no longer feasible in the case $n < p$ studied here, and we also demonstrate that a number of classical debiasing procedures become inconsistent. This challenge motivates our development and analysis of a novel procedure: we establish that it is consistent for the population mean under proportional asymptotics allowing for $n < p$, and also provide confidence intervals for the linear model coefficients. Providing such guarantees in the inconsistency regime requires a new debiasing approach that combines penalized M-estimates of both the outcome and propensity/missingness models in a non-standard way.