Riders need four qualifying scores to be eligible for Rio, of which two must average 70 per cent or more.
All that follows winning the Horse of the Year title in Hawkes Bay, so Thompson is clearly on a roll.
He's 26, born into horse sport and a total enthusiast. His family started the prominent Rich Hill Stud near Matamata, whose stallion Pentire sired last year's Melbourne Cup winner, the 100/1 shot Prince of Penzance. He teaches dressage around the country. Born in Hamilton, Thompson moved to Sydney with his family at a young age, when his brother Charles was accepted to the Australian Ballet Company.
There's a similarity with what they do. As Hamilton puts it, dressage is often known as ballet on horseback. It's also, for the uninitiated, a complicated ritual of movements the horse and rider must complete. It's about demonstrating the total compatibility and trust between the two.
Thompson describes dressage as "the key to any sport you want to do within equestrian".
He moved back to New Zealand with his girlfriend Holly a couple of years ago.
There's no tension between the two contenders either, Thompson insists. He would love to win the place but equally "if Julie happened to get the nod I would be very happy for her.
"The best thing about this is two people have qualified for the Olympics."
Antonello is in good shape too after a brush with death four months ago. The 13-year-old needed surgery for colic.
The plan now is to compete in Sydney at the final major event in Australasia starting on April 6, then fine-tune preparations, all the while crossing fingers for the New Zealand dressage selectors' decision.
That must be made by July. The pair have competed against each other 11 times, by Thompson's reckoning. He's beaten Brougham six times; been beaten five. It's that tight. "Neck and neck", he calls it.
Thompson is in this for the long haul. That means Tokyo 2020 and onwards from there. Brougham, 30 years older, may have a more limited time frame. That shouldn't have a bearing on the decision.
If Thompson misses out he insists the fact he has fully qualified for his country at the Games is still special "and that can never be taken off me. I'm just very proud of that achievement".
So after Sydney he'll keep busy teaching, keep Antonello in good shape - "and if all goes well we'll pack up and go off and live a dream".