Differentiability conjecture for mixed transport-diffusion quantum graph semigroups

Let $athsf{E}_t\not=\emptyset$ be the set of transport edges and $athsf{E}_d\not=\emptyset$ the set of diffusive edges. Let $A_{P,L}$ generate the semigroup $T(t)=e^{tA_{P,L}}$ on the state space $X$, with boundary conditions determined by $P$ and $L$. Let $T$ denote the length of the longest path contained in $athsf{E}_t$, computed while taking into account possibly coupled boundary conditions. For transport-supported initial data, the semigroup shifts the profile without smoothing until it reaches an endpoint.

Differentiability conjecture. If $athsf{E}_d\not=\emptyset$, then the semigroup is differentiable for every $t>T$; equivalently, for every $x\in X$, the map

$$(T,\infty)\ni t\longmapsto T(t)x\in X$$

is differentiable.

The transport component prevents immediate smoothing, analyticity, and immediate differentiability before profiles reach diffusive edges. The conjecture predicts that the diffusive part smooths the semigroup after the maximal transport-path length, even when the boundary conditions couple transport and diffusion components.