  In this paper, we show the regularity and uniqueness of the twisted conical
K\"ahler-Ricci flow running from a positive closed current with zero Lelong
number, which extends the regularizing property of the smooth twisted
K\"ahler-Ricci flow, known as Guedj-Zeriahi's existence theorem and Di
Nezza-Lu's uniqueness theorem, to the conical singularity case.
