Dense plastic subgroup in the ℓ1-plane

Determine whether the real plane ℝ×ℝ equipped with the ℓ1-norm ‖(x,y)‖1 = |x| + |y| contains a dense plastic subgroup.

Background

Theorem 3 establishes the existence of a plastically rigid dense subgroup in every strictly convex separable metric group. This applies to the Euclidean plane with the ℓ2-norm since it is strictly convex.

However, the ℓ1-normed plane is not strictly convex, so Theorem 3 does not apply, prompting the explicit question of whether a dense plastic subgroup exists in that setting.