We propose the AKRON-Kalman filter for the problem of inferring sparse dynamic networks from a noisy undersampled set of measurements. Unlike the Lasso-Kalman filter, which uses regularization with the l1-norm to find an approximate sparse solution, the AKRON-Kalman tracker uses the l1 approximation to find the location of a 'sufficient number' of zero entries that guarantees the existence of the optimal sparsest solution. This sufficient number of zeros can be shown to be exactly equal to the dimension of the kernel of an under-determined system. The AKRON-Kalman tracker then iteratively refines this solution of the l1 problem by ensuring that the observed reconstruction error does not exceed the measurement noise level. The AKRON solution is sparser, by construction, than the Lasso solution while the Kalman tracking ensures that all past observations are taken into account to estimate the network in any given stage. The AKRON-Kalman tracker is applied to the inference of the time-varying wing-muscle genetic regulatory network of the Drosophila Melanogaster (fruit fly) during the embryonic, larval, pupal and adulthood phases. Unlike all previous approaches, the proposed AKRON-Kalman was able to recover all reportedly known interactions in the Flybase dataset.