Sources familiar with the situation tell TMZ ... the family recently extended an olive branch to Mama June. They told her she can finally get her youngest daughter back, but with 2 big strings attached: dump Geno AND seek professional help.

We're told June said hell no to both.

As we first reported ... she looked worse for wear during a lunch outing with Geno, Monday in Hampton, GA. They both looked disheveled, and June still hadn't fixed her shattered veneers.

You'll recall, Geno drove his truck into the home he and June share a couple months ago, and was so wasted he couldn't walk.

Prior to that ... June's family staged an intervention that failed, and left Alana in tears.

Our sources say the family believes June's appearance Monday shows how desperate the situation is, but her refusal to get help -- along with Geno's presence -- is why Alana continues to live with her sister, Pumpkin.

We're told Alana's in a stable place now, despite her recent antics on social media ... when she pretended to snort cocaine. The family even took a recent trip -- sans June, of course -- to L.A. for vacay, and while they want their matriarch back, they want her to get help first.