Non-constructibility conjecture for common points of investigated conics

Let $x'$ and $y'$ be nonzero, and let $a_1,a_2,a_3$ be parameters. Define

$$a:=a_1^2-a_2^2,\qquad b:=a_2^2-a_3^2.$$

The investigated conics can be chosen so that their common point is not obtainable by a planar construction.

Non-constructibility conjecture. If $x',y'\ne 0$, then one can choose the parameters $a$ and $b$ above so that the common point of the investigated conics does not admit a planar construction.

This asserts the existence of parameter choices obstructing a planar construction of the common point. The supplied text does not provide an independent resolution of this stated claim; the status is therefore recorded as open.