Constitution Hill will school in midweek as connections ponder their next move with the nine-year-old.
Nicky Henderson's star made a sparkling winning Flat debut at Southwell on Friday night, winning by nine-and-a-half lengths and earning a Timeform rating of 106P.
He holds an entry in next month's Unibet Champion Hurdle but connections have yet to decide whether he'll take that up after falling three times in his last four National Hunt starts, including in the Fighting Fifth on his reappearance.
Speaking at Kempton on Saturday, the trainer said: "We will have one more school next week with Yogi Breisner. All that was going really well, but he didn’t have one last week as he was going to come in Tuesday, but I wasn’t sure I dared to do that with him running on Friday.
"If he had come out with an over-reach or something then he couldn’t have gone to Southwell."
And Henderson admits where they go now is a very difficult call to make.
"It is a seriously difficult decision. I know what Nico (de Boinville) would do and I know what Oisin (Murphy) would do, but it is not their call. It depends which side of the bed you get out of as one morning I get out and say ‘Come on, you’ve got to go and do this’ or 'don’t be so bloody stupid'. It is one of the two.
"I think I can safely say that he and everybody did a hell of a lot for the sport last night. It was unbelievable. It has been staggering coming here today that everybody has been on about last night. Well done to the guys who put it on as we asked for a race and they delivered it and they got what they deserved, and the horse did too.
"We have got to bear this cross of responsibility and we owe that to racing, and everybody. A lot of the people would love to see him there (Cheltenham) and do it, but there is also the faction that says no you shouldn’t. We have now got to really think it out."
Change made to Constitution Hill jumping technique
And connections have been working hard on Constitution Hill's hurdling technique.
"The schooling session will tell us a lot. He has definitely changed in his technique. He (Yogi Breisner) has got him putting a stride in. We are putting them in the whole time and he has been really good. I’m there to advise, but Yogi knows all about these things as he is the expert at it.
"He has said to me the great thing he has is Nico, as he is one of the best horsemen he has ever worked with. He loves doing it with him as he is so good, and so is the horse. We would like to do that again and that will be in the later part of the week. He has been teaching Nico and it is minute details. It is fascinating.
"I would say I would do it in public, but Yogi doesn’t do it in public. I’m not hiding anything, and I’m sure it will be filmed.
"There will be discussions with all sorts of people. I appreciate that a decision has got to be made sooner or later for the good of everybody, but there is still a bit of time to go.
"In one pocket I’ve got all the people who say come on get on with it and go jumping, and in the other pocket I’ve got people saying don’t and most of them are in the middle which is ‘I’ve not got a frigging clue!’"
Henderson admits he was blown away by the experience, performance and reaction to the Southwell success.
"It was terrific last night. It was unreal to be honest with you. You wouldn’t have dared think it was going to happen. It shocked us a bit. It has probably even muddied the waters as well.
"He has a future as a Flat race horse, which is a strange thing to be saying of a nine-year-old, but he does," he said.
Constitution Hill wins at Southwell
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