Dear Evelyn,
Thank you for your question, and for patiently awaiting my response. After contacting a textiles conservation specialist, I have a couple of recommendations for you. Hopefully some of them will be suitable to the nature of your site.
It is great that you have already identified, isolated, and sent to the freezer the hooked mats in worst condition. However, it would be ideal in terms of pest control to isolate all of the pieces, at least by bagging them, and eventually setting up a freezer project to kill any larva that may be about to hatch. I don't know the possibilities/limitations in your institution, specially being a site that comprises spaces with varied purposes, but museumpests.net is a great resource offering a range of plans to address active infestation of collections. You could evaluate what would be in line with the reality of Kings Landing, and implement a protocol following the recommendations/solutions available online.
Once you have addressed the infestation, storing these flat would be a good idea, as they are probably much more fragile than they seem before evaluating the extent of moth damage. An on-site consultation with a textile conservator could be very useful to inform how you decide to approach the stabilization/treatment of this collection. The American Institute for Conservation's Find-A-Conservator tool is an excellent way to locate a textiles specialist by area. Perhaps the Canadian Conservation Institute can also provide similar guidance though the professional conservation services section in their webpage.
The specialist that I contacted mentioned that if pieces like yours are very damaged, the yarns may actually be eaten through at the base of the hooked loops. Therefore, vacuuming can remove a lot of fiber, as they are only held in place by habit (which sounds like what you are experiencing). However, if you can't find a conservator to perform the stabilization treatment and take over the frass removal process, using a variable suction, HEPA-filter vacuum with a fine net over the head of the vacuum cleaner brush would be advised. The fine net over the head of the vacuum's brush will prevent bigger fibers from being removed, and the variable suction provides further control over the process. If you can't purchase a vacuum with the variable suction feature, you can modify a regular vacuum cleaner to make it have less suction by drilling holes in the tube (although this would mean that you have to think of what might come out the tube if working in an enclosed space).
I hope that some of these recommendations are practical in your circumstance. Let me now if you have any follow-up questions.
Best,
María Cristina
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María Cristina Rivera Ramos
Assistant Paper Conservator for Asian Art
Art Institute of Chicago
rivera.macristina@gmail.com------------------------------