Consider the moduli space $M_d$ ($d\geq 2$) of relatively minimal Jacobian elliptic surfaces $S\to\mathbb P^1$ with $p_g(S)=d$. This was constructed by Miranda (1981) using Weierstrass form and the techniques of Geometric Invariant Theory.

Let $R\subset M_d$ be the divisor consisting of surfaces with a reducible fiber. Let $W$ be the locus of surfaces with non-trivial Mordell-Weil group. Cox proved (1990) that $W\cap (M_d-R)$ is a countable union of codimension $d$ subvarieties.

My question is about $W\cap R\subset R$. Is this known to be a proper inclusion? Do we have any examples of (non-K3) elliptic surfaces with reducible fibers and trivial Mordell-Weil group?