Subdivision-preservation conjecture for maderian digraphs

Let $F$ be a digraph. A subdivision of $F$ is obtained by replacing each arc with a directed path, with the replacement paths internally vertex-disjoint. Say that $F$ is $\delta^+$-maderian if sufficiently large minimum out-degree forces a subdivision of $F$.

Subdivision-preservation conjecture. If a digraph $F$ is $\delta^+$-maderian, all subdivisions of $F$ are $\delta^+$-maderian as well.

The statement is presented as a plausible preservation principle and remains open in the supplied text.