Cardinal characteristics for Menger-bounded subgroups

Machura, Shelah and Tsaban showed in [M. Machura, S. Shelah, B. Tsaban, Squares of Menger-bounded groups, Trans. Amer. Math. Soc., in press, http://arxiv.org/pdf/math.GN/0611353, 2007] that under the condition, that a relative d^′(P) of the dominating number is at least d, there are subgroups of the Baer-Specker group whose kth power is Menger-bounded and whose (k+1)st power is not. We show that the sufficient condition implies r?d and indeed can be replaced by r?d. This result includes an affirmative answer to a question by Tsaban on a possibly weaker still sufficient condition. We show that it is consistent relative to ZFC that g?r<d and there are subgroups of the Baer-Specker group whose kth power is Menger-bounded and whose (k+1)st power is not.