Enter the alcove and perform the Chimney Jump to grab the ledge above; pull up. The two switches here raise the gates below, so you can climb into the alcoves exactly above. However, they are timed and close after about 30 seconds. First, use the right hand switch (if the entrance is to your back). Then use the left one.

Drop to the floor below and head to the right hand gate (as you face the entrance). Jump up to grab the first handhold on the gate.

Then jump to grab the next handhold above. Traverse to the very left and jump to grab the handhold on the left wall which is above the entrance. Now you can relax and take your time. Shimmy to the left.

When you reach the end, jump up to grab the handhold above.

Now lean back (make Lara look backwards) and use the Right Stick to move the camera, so you can see the left hand gate. When you see it lowering (you will also hear a crashing sound), jump to grab the edge of the alcove above.

Pull up and roll into the low opening at the end. Get the Treasure (5/28) and return back to the edge of the alcove.

[Note - To get the Treasure, you just need the right hand gate open (facing the entrance) and this only for a few seconds, so as to reach the handhold on the wall above the entrance. Once there, do not rush and just take your time to reach the alcove with the Treasure. There are many players who follow a more complicated method by rushing to the left gate, jumping back to the right one, then again jumping back to the left one. This is not necessary.]