Idea for reducing crossover distortion in class AB amplifier

I recently was looking at class AB amplifiers and had an 'idea' of how to reduce crossover distortion. The idea is to prevent Q1 from completely turning off by drawing a constant current I1 of around 1mA. Conversely I2 keeps Q2 from turning off completely. The value of the current source would be a compromise between losses and distortion reduction.
This circuit is only conceptual, it may need additional blocking diodes to protect the current sources. Nevertheless, if the transistors are kept just conducting, i.e. slightly on, then the crossover distortion would be reduced.
I'm sure I'm not the first to come up with this. What do people familiar with class AB design think of this?

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Kirchhoff's law suffers no exception, and the current sources will neutralize each other at all times, meaning the circuit with the current sources is exactly equivalent to the circuit without them.
To achieve your goal, you would need something like steering diodes, but in practice it will be complicated and the problems will be difficult to solve
 
@Elvee I'm sure if it were simple, someone would have already done it. Yes, the current sources cancel out in the simplified circuit diagram. There would definitely need some addition steering/protection diodes. I'm wondering if this has been attempted before in a similar shape or form?