As I said I can't help you. I don't have the mathematical or phsyical knowledge. I only suggest that, when you argue it out with them, you stick to mathematics and physics, and not constantly look for evidence that they "spilled the beans".
Well, it's not a physics or math argument really. Spilling the beans is my way of saying how I understand their goal in using the PCA technique. It's probably not technically incorrect, but the assumption in Fig 5 is obvious, they want proxy data to match the instrumental record. They found that data: bristlecone pine (from climateaudit website). Now I have to figure out two problems: whether they applied the PCA method correctly (likely) and whether it is correct to "pull" a temperature signal from noisy, low resolution proxy data using PCA (doubtful). The reason I say doubtful is there are more direct ways to determine the temperature component of a proxy (such as compensating for rainfall). The PCA method ignores that extra information. That's not a fatal flaw but it does raise some questions about its applicability.