Conjecture on products and base change of $l$-$i$-modular extensions

Let $\Omega/k$ be a $q$-finite extension, and let $K$ and $L$ be intermediate fields. An extension is called $l$-$i$-modular when it has the modularity property described in the preceding definitions.

Product and ascending base-change conjecture. (1) The compositum of two $l$-$i$-modular extensions is $l$-$i$-modular: if $K/k$ and $L/k$ are $l$-$i$-modular, then $K(L)/k$ is $l$-$i$-modular. (2) $l$-$i$-modularity is preserved when the base field is changed in the ascending direction; in particular, $L(K)/L$ is $l$-$i$-modular if $K/k$ is $l$-$i$-modular.

The conjecture extends the preceding closure results for $l$-$1$-modular extensions to arbitrary $i$. The supplied text presents it as an expected generalization, and provides no resolution.