What is the exact density of 3x-1 obstruction residues?

For each $L$, define $O_L$ as follows. For an odd integer $r>1$ with $L>v=v_2(r-1)$, set $m=(r-1)/2^v$, start $(a_K,b_K)=(r,2^L)$, $(a_I,b_I)=(m,2^{L-v})$, and $(c,d)=(2^{-v},-2^{-v})$. At each step set $A_K=3a_K-1$, $\alpha=v_2(A_K)$, $A_I=3a_I-1$, and $\beta=v_2(A_I)$. Stop if $\alpha\ge v_2(b_K)$ or $\beta\ge v_2(b_I)$. Otherwise replace $c$ by $c2^{\alpha-\beta}$, replace $d$ by $(3d+c_{\mathrm{old}}-1)/2^\beta$, replace each $a$ by the matching $A$ divided by its power of 2, and replace each $b$ by $3b$ divided by that power. Put $r\in O_L$ exactly when $3d+c=1$ at the stopping step. The ratios $|O_L|/2^L$ are nondecreasing and have a limit $c_W$. Determine $c_W$ exactly.