How to check that the HDDs are spinned down?

You'll need to download your logs (status > logs > download all logs), extract the zip contents and look in daemon.log. There will be entries in there regarding the disks "spinning" if the disks are spinning down and up.

The HDDs and the sytem are able to spindown:

Jan  4 21:27:18 ReadyNAS noflushd[1830]: Spinning down disks.
Jan  4 21:38:04 ReadyNAS noflushd[1830]: Disks spinning up after 10 minutes.
Jan  4 22:03:58 ReadyNAS noflushd[1830]: Spinning down disks.
Jan  4 22:55:09 ReadyNAS noflushd[1830]: Disks spinning up after 51 minutes.
Jan  4 23:25:40 ReadyNAS noflushd[1830]: Spinning down disks.
Jan  5 00:15:36 ReadyNAS noflushd[1830]: Disks spinning up after 49 minutes.
Jan  5 00:36:31 ReadyNAS noflushd[1830]: Spinning down disks.

But the last entry is from January 6. I think there is a process in the background which keeps them runnning :cry:
Which of all the log-files could help us to find out who the bad guy is?

It's me again. I had all PCs an network stuff shutdown. The deamon log is:

Jan 19 18:36:38 ReadyNAS ntfs-3g[21439]: Unmounting /dev/sdc1 (core) 
Jan 22 18:06:27 ReadyNAS chronyd[2977]: Can't synchronise: no majority
Jan 22 18:10:44 ReadyNAS chronyd[2977]: Selected source 64.94.70.147

So no spin-down :cry:
What can I do??