Julius' sister, Antoinette Jones, tells TMZ ... Kim has been in touch with the family in recent days to see how they're holding up -- but more importantly, to get a pulse on how JJ himself is handling the dire situation.

She's even posted Instagram updates of late, shedding light on where things stand with Julius' case ... and, again, calling on Oklahoma's Gov. Kevin Stitt to do the right thing.

While it may come off as mere lip-service for some ... Antoinette tells us Kim's lobbying and advocating on her brother's behalf has meant the world to their family behind the scenes -- insisting she's a "real one" and actually cares about what happens here.

Not just that ... but Antoinette says Kim knows her stuff too, legally speaking -- and she's impressed the Joneses by her breadth of knowledge on the facts of the case and beyond.

Most influential, though, is Kim's massive cultural influence ... which Antoinette tells us has been immeasurable in putting a spotlight on Julius' story -- and which she says she and the rest of the fam are beyond grateful for ... regardless of what happens on Thursday when Julius is scheduled to be executed.

Quick note about that ... the Parole Board has recommended Julius should not be executed but rather serve out a life sentence behind bars ... but the ball has been in Governor Stitt's court with no movement ... and now, it's almost at the point of no return.

Considering the Gov. can still grant clemency in the 11th hour ... Antoinette is asking anybody with a following and some pull to lean in and help them get the word out about what they call a travesty of justice.

As you may know by now ... Julius was charged and convicted for first-degree murder way back in 2002 -- with prosecutors pinning him to the shooting death of 45-year-old Paul Howell. Julius has always maintained his innocence ... and much has been made about weak evidence in the case linking him to the crime, not to mention racially charged undertones.