On a first hit distribution of the running maximum of Brownian motion

Considering the running maximum $S_t$ of a standard Brownian motion $B_t$, we study the distribution of the first time $T$ when it becomes less than the running time $t$. The motivation for our work comes from a mathematical model for animal foraging. We also compute the joint distribution of $T$ and $B_T$.
Reference: Randon-Furling, Salminen, Vallois. "On a first hit distribution of the running maximum of Brownian motion." Stochastic Processes and their Applications (2022)
https://www.sciencedirect.com/science/article/pii/S0304414921002258