I have now managed to cache css, js and other files. It turned out that ASM and AAM do not play well together. I will look into if I can have both on at the same time. But now I face a new problem:
V2: X-WA-Info Format Version
S10206: Response was served from the origin web server, because the content was uncacheable.
A80364: Application: /Common/aam_svartboks_cache
P37379: Local-policy: /Common/Wordpress
N13710: Request Policy Node: Pages
RN0: Response match did not supersede request match
UCI hash: 0
Object type: html
Object group: pages
Request served from TMM: 0.1
Request owned by TMM: 0.0
Entity hit count (local/remote): 0/0
Document hit count (local/remote): 0/0
Document not cacheable (negative cache entry).
Reason: Response validators (ETag, expiration, or Last-Modified) prevent caching.
Bypass: Content received is not cachable.
Parking: Not parked.
There is no ETag, Expiration or Last-Modified sent from the web server...
Nimbostratus