As long as you use a neutral / low ph prespray and follow up with an acid rinse or a neutral extraction detergent you should be fine. In the past I have had good results with Craftex Pre enzyme solution extracted with Chemspec Optimal Rinse. In recent times I have had good results using Restoremate Bio Clean extracted with Hydramaster Clearwater rinse. Shampoo extracted with Fib and Fab would work well. I use those acid extraction solutions not to neutralize, but because they have added cleaning power without raising the ph level. Most acid rinses are just acid rinses, and have no cleaning power at all.
The sapphire hand tool does help control the moisture level as good as can be, however linen does retain alot of moisture so air movers are essential.