Linear equations do not assure positive solutios, but negative capacities would imply very costly active electronic elements, rendering the proposal ineffective.
The selectors are explained in the app, try to move them until you find that the capacitor (point C1) has a negative value (this is highlighted by the vertical velvet line changing its direction)
The colored curves explain the essence of the compensation: The charge curve (brown) and the opposite of the compensator curve (red) meet at the operating frequencies (W1 and W2). This implies that their net effect vanishes, thus leaving all the intensities without non-active components.
It is possible to re-arrange the solution in order to get only positive capacities, using an additional parallel capacity (green colour point C3) in parallel to the rest of the shunt branches.
You can find an acceptable solution rising the green point C3 until all three black rays point upwards. This implies that all the compensating capacities (C0, C1 and C2) are positive.
Capacities Cj are linked with inductances Lj by their common frequency of resonance fj, which is always positive. In this way, a positive Cj implies also a positive Lj.
The problem is that manual solutions are not desirable. It is preferred that the compensation solution arises mmdirectly without human intervention.
A first step in order to make this process in an authomatic way would be proposing a value for C that is so high that it leave no option for any Cj to be negative.
The problem now is that it implies the use of very high values for the compensator capacity. Condensers with so high capacities are very expensive, and for this reason they should be avoided.