The stronger-than relation conjecture for graphs without semi-edges

Let $A$ and $B$ be connected graphs. Write $A\triangleright B$ when every simple graph that covers $A$ also covers $B$, and write $A\longrightarrow B$ when $A$ covers $B$. A graph has no semi-edges if none of its edges is a semi-edge.

The stronger-than relation conjecture. If $A$ has no semi-edges, then

$$A\triangleright B\quad\Longleftrightarrow\quad A\longrightarrow B.$$

This conjecture characterizes the stronger-than relation for graphs without semi-edges and would show that the apparently broader relation yields no additional comparisons in that case. It was formulated as an open conjecture in the GROW 2022 open problem session.