What is the relation between airspeed and altitude at fixed throttle setting in a private jet?
The basic principles needed for an answer are:


Thrust varies linearly with air density.
Lift at constant angle of attack (AoA) equally varies linearly with air density.
Increasing the angle of attack from a low value will improve the lift to drag ratio (L/D), while increasing it from a higher value than the AoA at optimum L/D will decrease it.
The aircraft will settle at the point where speed equals drag and lift equals weight. If the altitude change improves L/D, the aircraft has excess thrust to fly faster.
Due to the atmospheric lapse rate, the speed of sound decreases with altitude.
Above a certain Mach number ($M_{dd}$ = drag divergence Mach number), drag increases nonlinearly over dynamic pressure.


Now the answer depends on where you start:


Low altitude, moderate throttle setting: The aircraft will speed up, because it was flying at a low AoA in dense air. Climbing brings it closer to the polar point of optimum L/D.
Low altitude, high throttle setting: If the aircraft flew at close to its $M_{dd}$, climbing will put it in colder air where it needs to fly slower to maintain the same Mach number. Depending on the starting speed, the aircraft might slow down.
High altitude, moderate throttle setting: Now the thrust decrease with altitude will dominate the equation, because the aircraft will fly close to its optimum polar point. Once it flies above the AoA of minimum drag, climbing will slow it down.
High altitude, high throttle setting: If it flies above the tropopause and close to the polar point of minimum drag, the air temperature does not change with altitude, and climbing will not change flight speed. It will stay close to its $M_{dd}$. Once it flies above the polar point of minimum drag, however, climbing will again slow it down.