Random-selection analogue of the alternating-walker limit theorem

Consider the two-walker model in which the walker to move is selected randomly, and let $M^{\rm rd}_\infty$ denote the limiting proportion of the left edge weight. The alternating-walker result comprises a central limit theorem and the assertion that the limiting variable has no atoms. Random-selection analogue conjecture. The corresponding central limit theorem and no-atoms property should also hold when the moving walker is selected randomly. This is proposed as an analogue of the proved alternating-walker theorem; the random-selection case remains open in the supplied text.