The six-year-old is the 1-5 market leader for the Festival highlight in March after another wide-margin success in the Christmas Hurdle at Kempton on Boxing Day, routing stablemate Epatante by 17 lengths - an increase of five lengths on his Fighting Fifth verdict over the crack mare.
Henderson had mentioned the Champion Hurdle Trial at Haydock on January 21 as a possible stopping point for his star hurdler, while the Irish Champion Hurdle takes place at Leopardstown on February 5. But neither option looks likely now.
The Seven Barrows handler would have been keen to run at Cheltenham had the International Hurdle that was originally scheduled for December 10 been resurrected, but that is not a possibility, despite Henderson raising the idea.
He told Nick Luck's Daily Podcast: "We've ummed and ahhed a fair bit, I did have what I thought was quite a bright idea but it doesn't quite come off and I can understand why, but I did sort of enquire whether we might have the International Hurdle that was abandoned rescheduled on Trials Day at Cheltenham.
"If that had been possible I would have gone there with him, but I really don't like the idea of Haydock and so I think almost certainly Constitution Hill will not run again before Cheltenham.
"He's obviously very good fresh, he was very good the first time this year in the Fighting Fifth. Last year we went from the Tolworth to Cheltenham and you couldn't have had him any better so I really don't see any point, there's nothing to be gained.
"I might have an open day or something so people can come and see him - I would have thought that was a possibility."
Epatante won the Champion Hurdle in 2020 and finished third and second to Honeysuckle in subsequent renewals, but having been comprehensively beaten twice by Constitution Hill, her future plans have yet to be decided - with similar comments applied to Relkeel Hurdle third First Street.
Henderson added: "We haven't discussed (Cheltenham) with Epatante yet, we will see what she does. First Street is a difficult one - I mean where does he go? He's on the verge of it, but he didn't stay two and a half on that ground on the weekend. You would enter him anyway. He's a very decent horse."
Seven Barrows houses the Relkeel victor Marie's Rock, who is on course to defend her Mares' Hurdle title at the Festival before a potential step up to three miles in the Liverpool Hurdle at Aintree.
Henderson said: "What I would really like to do with her is go to Cheltenham, I would love her to defend her title, and then I would seriously think about going to Aintree and trying her over three.
"We've always talked about it and everything did go swimmingly well, you find Constitution Hill coming into the the two-and-a-half-mile at Aintree.
"Are we tempted to put her in the Stayers' (at Cheltenham)? The answer is probably no, we would stick to the Mares' and then (look at Aintree) not just because of Constitution Hill, but because we've genuinely always discussed whether she would stay three."
Jonbon was 22 lengths behind Constitution Hill when second in the Supreme last season, but has made a seamless switch to fences this campaign, cruising to victory on his chasing bow at Warwick before sealing a ready eight-length verdict upped to Grade One level at Sandown at the beginning of December.
A Warwick return on February 11 is now his agenda.
Henderson said: "Jonbon will go to the Kingmaker at Warwick which is the only race he can run in really. He's been round there once and it's a good jumping test - I think that would do him good."
Former two-mile star Shishkin is set to step up in trip at Kempton later this month after a disappointing return in the Tingle Creek, with Henderson not even entering him for the Champion Chase at Cheltenham, instead opting to put him in the two-and-a-half-mile Ryanair Chase, plus the three-and-a-quarter-mile Cheltenham Gold Cup.
He added: "The plan at the moment is to go to Kempton on Saturday week for the Silviniaco Conti over two and a half miles."