Parameterized form of Zilber's conjecture on intersections with tori

Let $K$ be an algebraically closed field extending $k_0$, and let $B=\mathbb{B}(K)$ be a semiabelian variety defined over a field $k_0$ of characteristic zero. For a subvariety $W(x,y)$ of $B^{1+k}$, a point $c\in B^k$, and a coset $\alpha+C$ of a proper algebraic subgroup $C$ of $B$, let $S$ be an atypical component of the intersection of $W(x,c)$ and $\alpha+C$. Parameterized CIT. For every $k\geq 0$, every such $W(x,y)$ defined over $k_0$, and every $c\in B^k$, there exist a finite collection of proper algebraic subgroups $C_1,\dots,C_s$ of $B$ and elements $\alpha^1,\dots,\alpha^r\in B$ such that, for every coset $\alpha+C$, some $i\in\{1,\dots,s\}$ and $j\in\{1,\dots,r\}$ satisfy that $S$ is contained in $\alpha^j+C_i$ and is a typical component of the intersection of $W(x,c)$ and $\alpha+C$ with respect to $\alpha^j+C_i$. The source states that the original CIT implies this parameterized version in the multiplicative-group case; it gives no resolution status.