 Solution

Let $T$ be the support of a vector $\hat{x}$. If the columns indexed by $T$ were linearly dependent, there would be a nonzero $h$ supported in $T$ with $Ah=0$. For sufficiently small real $t$, the nonzero signs of ${\hat{x}}_T$ remain fixed, soBoth positive and negative $t$ preserve the residual and hence feasibility, even with noise. A nonzero slope contradicts minimality in one direction. A zero slope gives distinct minimizers, contradicting uniqueness. Thus those columns are linearly independent, andThis proves the sparse vector assertion without assuming a noiseless residual.