To complete this, you need to open the door with the Music Box Cylinder. This requires you to obtain and use the following items: Sculpture Gear, Dual Pistols, Empty Bucket, Sundial Gnomos, Grapple, Wrench, Decorative Arrow, Decorative Bow, Music Box Cylinder.
You do NOT need to obtain Lara's Journal or enter Lara's bedroom area to complete this, and you do not need to do anything that accesses artifact areas (some buttons in the training area, the crossed spears button, the Atlas button, etc.)
Here's my suggestion for the progression:
1. Climb the boxes in the hall, get the Sculpture Gear.
2. Drag the "fragile" box onto the pressure plate, revealing the Decorative Arrow.
3. Go upstairs, left, first door. Follow to room with chandelier.
4. Go upstairs, far wall, push books into each of the bookcase walls. Pistol storage opens. (Take pistols.)
5. In preceding room, jump onto picture frame; slides down to reveal target. Jump back onto balcony, shoot target, secret door opens on first floor.
6. Go into secret passage. Pick up Empty Bucket.
7. Proceed to end of secret passage, pull level, enter room. Go upstairs, enter room, shoot rightmost display case, take Sundial Gnomos.
8. Return to preceding room. Enter security code, exit to main hall.
9. Proceed to garden area. (Facing stairway, door on left.) Proceed to sundial. Install Sundial Gnomos. Align sundial to 11, then 2, then 7. Gate opens.
10. Proceed to central area. Suggest turning right at locked gate, and hug the left wall at all times. Take Grapple.
11. Grapple gearbox cover, install Sculpture Gear.
12. Grapple gate open. Return to main hall.
13. Proceed to training room. (Facing stairway, door on right.) Go to pad labeled "1". Climb to button, press button.
14. Go to pad labeled "3". Climb up and to the right. DO NOT jump to swing pole. Climb to button on wall near entrance, press button.
15. Return to pad labeled "3". Again, up and to the right. Jump to newly available swing pole, make your way to ceiling scaffold. Take Wrench. Return to main hall.
16. Return to garden area, keep to right, proceed to blue equipment. Install Wrench. Turn wrench until 3 green lights. (Water service restored.)
17. Approach nearest fountain. Use Empty Bucket. Automatically will take Bucket of Water. Return to main hall.
18. Use Bucket of Water on fireplace. Take Decorative Arrow.
19. Proceed to construction area. (Main hall, door next to where the Sculpture Gear was found.) Move dumpster with pipes in between scaffolding.
20. Shoot rope attached to diagonal board at far end of room.
21. Climb onto boxes nearby, make your way to the first scaffolding you reach. Jump onto handle hanging from spear near entrance, spear pulls down.
22. Return to scaffolding, jump onto spear, swing to balcony. Shoot rope (1/3). Return to ground floor.
23. Return to scaffolding, jump to pipe dumpster, jump to second scaffolding. Use grapple to wall-walk to third scaffolding. (Recommend you push box onto floor, just in case you fall -- however, not necessary). Walk past Atlas, shoot rope (2/3).
24. Return to area where box was/is. Jump onto teetering board. Delay a fraction of a second, to allow board to tilt, then run to opposite side, jump up to scaffold wall. Make your way to central balcony, shoot rope (3/3). Statue falls through floor.
25. Jump into water, dive, take Decorative Bow. Surface, return to main hall.
26. Proceed to garden area. Install Decorative Bow, Decorative Arrow. Pull lever. Retrieve Music Box Cylinder. Return to main hall.
27. Proceed upstairs, right, 2nd door. Follow to locked door, install Music Box Cylinder.
28. Enter door. [finish]