The non-primitivity conjecture for random countable Boolean algebras

Let $W$ be the Cantor set $\mathscr{D}_{1}$ and let $R=\operatorname{Co}(W)$ be the countable atomless Boolean algebra. Let $I$ be the set of finite binary strings, including the empty string, and let $\{A_i\in R\mid i\in I\}$ be a binary tree of non-zero elements of $R$ generating $R$, with $A_{\emptyset}=R$ and $A_i=A_{i0}\dotplus A_{i1}$. Define a measure $\sigma\colon R\rightarrow\{0,1,o\}$ by $\sigma(0)=o$, $\sigma(R)=1$, by assigning $0$ to both children of any $A_i$ with $\sigma(A_i)=0$, and by assigning $1$ to $A_{i1}$ and assigning $0$ or $1$ to $A_{i0}$ with probability $0.5$ whenever $\sigma(A_i)=1$. Non-primitivity conjecture. For the measure $\sigma$ constructed above, $K(W)$ is $\sigma$-primitive with probability $0$. This suggests that almost all countable Boolean algebras are non-primitive, whereas most easily constructed such algebras are primitive.