Identify the distinguished point within the block {2,13,15} of T

Identify the point {2} intrinsically from the block {2,13,15} in the 15-point Steiner triple system T.

Background

In Step 1 of the construction, the authors use a 15-point Steiner triple system T whose automorphism group is isomorphic to S_3. The block {2,13,15} can be identified within T, but the three points in that block are fully symmetric: every permutation of {2,13,15} extends to an automorphism of T. Consequently, the construction can canonically recognize the block but cannot distinguish which of its points corresponds to {2}. The unresolved issue is therefore whether, or by what additional structure, the point {2} can be intrinsically identified from that block.