Recurrence of the balanced excited random walk M(2,1,1)

Prove that the balanced excited random walk M(2,1,1) is recurrent.

Background

The two-dimensional walks M(2,1,1) and M(2,1,2) are discussed as special cases of balanced excited random walks. Prior work on M(2,1,1) established upper and lower bounds for its range, including an upper bound on the expected range, but did not determine recurrence. The paper explicitly records recurrence of M(2,1,1) as unresolved, in contrast with its main result proving recurrence for M(2,1,2).