About the first log sent, we can't find any evidence of delays.
We just compared the timestamps of the log lines which report the new messages received from the Server and the timestamps included in the message payloads.
What we see it that the two timestamps are very similar, with a delay of much less than a second that is steady throughout all the log.

This means that any delays must occur after the reception of the updates.
After the reception, the updates are added to an "event thread" which invokes the application SubscriptionListener. So we have to suspect that the block or delay is inside the application listener.
To ascertain this, we need more log that, unfortunately, is not available in LS SDK library. You should add some logging inside your listener and in particular in the onItemUpdate callback to report each invocation and the data received.

Can you also provide information on your observations?
The log sent spans a time interval from 13:20 to 13:40. Could you report how was the observed delay between 13:20 and 13:40 and the behavior in time (for instance, did it increase with time)?

The second log was not received. I suspect that you sent it to Giuseppe's address rather than LS support address.

Anyway, if the problem is that delays are caused by the machine going into sleep mode, this should be clarified better.
Obviously, during sleep mode the application is not being supported by the system, so we don't provide any guarantee on that state.
Do you mean that, when resuming after the sleep mode, the application shows all the updates occurred during the sleep phase and that it is unable to catch up and keeps sending delayed data?