Three pipes are connected to an inverted cone, with its base at the top. Two inlet pipes, $A$ and $B$, can fill the empty cone in $8$ hours and $12$ hours, respectively. The outlet pipe $C$ (at the bottom) can empty a filled cone in $4$ hours. When the cone is completely filled with water, all three pipes are opened. Two of the pipes remain open for $20$ hours continuously and the third pipe remains open for a lesser time. As a result, the height of water comes down to $50%$. Which of the following options would be possible?