Retract-smoothness conjecture for Sidorenko graphs

Let $H$ be a Sidorenko graph, and let $T$ be a tree that is a retract of $H$, meaning that there is a graph homomorphism from $H$ to $T$ whose restriction to $T$ is the identity. Say that $T$ is smooth in $H$ when it satisfies the smoothness condition defined in the source.

Retract-smoothness conjecture. If $H$ is a Sidorenko graph and $T$ is a retract of $H$, then $T$ is smooth in $H$.

This conjecture links the graph-theoretic retract property to the analytic smoothness condition introduced in the paper. The source presents it as a natural conjecture and gives no resolution.