Ah Ha! I figured it out. You have to set the CSS 'position' of the nr-dashboard-text in the group to 'static'. Here is an example with two tabs showing the css that will only effect one of the tabs:
[{"id":"bcc301e2.996758","type":"ui_text","z":"4cd4f4f3.7effcc","group":"f03bcc4.355aa3","order":1,"width":"7","height":"1","name":"","label":"Alarm Status","format":"{{msg.payload}}","layout":"row-spread","x":390,"y":140,"wires":[]},{"id":"89ecd5f7.6159b8","type":"ui_text","z":"4cd4f4f3.7effcc","group":"f03bcc4.355aa3","order":2,"width":"7","height":"1","name":"","label":"Time Remaining","format":"{{msg.payload}}","layout":"row-spread","x":390,"y":220,"wires":[]},{"id":"cb4a4780.37f128","type":"inject","z":"4cd4f4f3.7effcc","name":"","topic":"","payload":"Disarmed","payloadType":"str","repeat":"","crontab":"","once":false,"onceDelay":0.1,"x":150,"y":140,"wires":[["bcc301e2.996758"]]},{"id":"f0411449.136a7","type":"inject","z":"4cd4f4f3.7effcc","name":"","topic":"","payload":"0","payloadType":"num","repeat":"","crontab":"","once":false,"onceDelay":0.1,"x":140,"y":220,"wires":[["89ecd5f7.6159b8"]]},{"id":"cd068ce9.16809","type":"ui_template","z":"4cd4f4f3.7effcc","group":"f03bcc4.355aa3","name":"","order":3,"width":0,"height":0,"format":"// note: the borders are just for visually \n// seeing what each element covers\n<style>\n#Alarms_Group_1 {\n xborder: solid 1px black;\n}\n#Alarms_Group_1_cards {\n xborder: solid 3px red ;\n height: 40px !important;\n}\n#Alarms_Group_1_cards .nr-dashboard-text {\n xbackground-color: blue;\n xborder: solid 5px green;\n height:20px !important;\n position: static;\n}\n</style>\n","storeOutMessages":true,"fwdInMessages":true,"templateScope":"global","x":260,"y":60,"wires":[[]]},{"id":"526a9f2.c56c66","type":"ui_text","z":"4cd4f4f3.7effcc","group":"1d5a4116.5c9cc7","order":1,"width":"7","height":"1","name":"Alarm Status 2","label":"Alarm Status 2","format":"{{msg.payload}}","layout":"row-spread","x":380,"y":320,"wires":[]},{"id":"4943a5ab.d5ba8c","type":"ui_text","z":"4cd4f4f3.7effcc","group":"1d5a4116.5c9cc7","order":2,"width":"7","height":"1","name":"Time Remaining 2","label":"Time Remaining 2","format":"{{msg.payload}}","layout":"row-spread","x":380,"y":400,"wires":[]},{"id":"3a464850.938ab8","type":"inject","z":"4cd4f4f3.7effcc","name":"","topic":"","payload":"Disarmed","payloadType":"str","repeat":"","crontab":"","once":false,"onceDelay":0.1,"x":130,"y":320,"wires":[["526a9f2.c56c66"]]},{"id":"a7f03dfe.20951","type":"inject","z":"4cd4f4f3.7effcc","name":"","topic":"","payload":"0","payloadType":"num","repeat":"","crontab":"","once":false,"onceDelay":0.1,"x":120,"y":400,"wires":[["4943a5ab.d5ba8c"]]},{"id":"f03bcc4.355aa3","type":"ui_group","z":"","name":"Group 1","tab":"abe23ae3.9bbb68","order":1,"disp":true,"width":"7","collapse":false},{"id":"1d5a4116.5c9cc7","type":"ui_group","z":"","name":"Group 2","tab":"eb8ce6b7.5818c","order":1,"disp":true,"width":"6","collapse":false},{"id":"abe23ae3.9bbb68","type":"ui_tab","z":"","name":"Alarms","icon":"dashboard","order":4},{"id":"eb8ce6b7.5818c","type":"ui_tab","z":"","name":"Alarm 2","icon":"dashboard","order":3}]
