Determine whether non-degenerate maps to the blow-up are automatically nice

Determine whether every non-degenerate map from a general curve $C$ to $\operatorname{Bl}_{\mathbb P^s}(\mathbb P^r)$ is nice, namely whether it satisfies $H^1(C,f^*T_{\operatorname{Bl}_{\mathbb P^s}(\mathbb P^r)})=0$.

Background

A map to the blow-up is called nice when it is non-degenerate and its pulled-back tangent bundle has vanishing first cohomology. Niceness guarantees that the corresponding moduli point is smooth of the expected dimension.

In characteristic zero, the moduli space of non-degenerate maps is known to be pure of expected dimension, but this does not imply that every map is unobstructed. The authors identify automatic niceness as an analogue of the Gieseker–Petri phenomenon for maps to the blow-up.