Squid Game II

$10$ contestants are arranged into a line on a bridge and in front of them lay ten left tiles and ten right tiles side by side. In order to cross the bridge, the contestants must cross $10$ tiles, and at each step, the person in front must pick either the left or right tile to step on. However, for each left & right tile pair, there is exactly one sturdy tile and one faulty tile, but the contestants cannot tell them apart. The contestants cross the bridge in their assigned order with the first person picking either the left or right tile, and continuing to lead unless either a faulty tile is picked (resulting in elimination) or person one reaches the other side. If the first person is eliminated before reaching the other side, the person second in line assumes the lead picking until he/she is eliminated (or reaches the other side), and so on. The winner of the game is the $\textit{first}$ person to reach the other side. Let $p_i$ be the probability that the $i$th contestant in the line wins. Find ${ \underset i \sup} \hspace{3pt} p_i$.