Block reduction conjecture for graph covers

Let $H'$ be a graph, and let $H$ be a block graph of $H'$, meaning a graph obtained from $H'$ by replacing each block according to the block construction used in the paper. The problem ${\sc H\text{-}Cover}$ asks whether a simple input graph admits a cover mapping to $H$.

Block reduction conjecture. The problem ${\sc H\text{-}Cover}$ for simple input graphs polynomially reduces to ${\sc H'\text{-}Cover}$ for simple input graphs.

The conjecture is proposed as a possible extension of the paper's reduction method and could help transfer complexity results between target graphs. The supplied context does not establish whether it has been proved or refuted.