I've called UA, and this time a 3rd agent has informed me that my connection is not Valid. She said that she will need some time to contact TG to ask them what needs to be done. .. So she has asked me to sit tight till then. But she said most likely I have to change my dates of departure.
Anyway, I pressed her on what is the LEGAL minimum connecting time. . . after much hesitation she told me 1 hr. 15 minutes.
Thanks All!