Does the Terminal Adapter have an inactivity timer?

Yes, it is controlled by S-registers S29 and S30.

S-Register: S29
Usage: S29=n; S29?
Function: On-line Inactivity Timer Period
Unit: 0.25 seconds
Range: n=0--does not disconnect
n=1-255 (0.25-63.75 seconds)
Default: 240 (60 seconds)
Description: S29 sets the on-line inactivity timer period and affects the on-line inactivity timer S30 by adjusting its unit of time. For example, this allows the inactivity of data calls to be timed down to 0.25 seconds and if no data is received within 0.25 seconds the data call will disconnect. This is useful for quick connect-disconnects or for continuous connect-disconnect cycles when combined with the use of Persistent DTR Dialing (PDD). By default the period is 60 seconds (1 minute).

S-Register: S30
Usage: S30=n; S30?
Function: On-line Inactivity Timer
Unit: (n * S29 * 0.25) seconds (S29=240 by default)
Range: n=0 (does not disconnect),
n=1-255 (1-255 minutes by default (S29=240))
Default: 0 (does not disconnect)
Description: S30 causes the TA to disconnect a data connection if no data is transmitted or received for the specified time. It will NOT cause an analog call (voice/modem/fax) to disconnect. The timer will restart any time a data character is passed through the serial port (either sent or received). The inactivity timer is disabled by setting S30=0 or S29=0. By default S30=0, so data calls are not disconnected if sitting idle.