What is the weight of a container filled with flying birds?

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SUMMARY

The total weight of a container filled with flying birds remains unchanged regardless of the birds' flight. In the scenario presented, the container has a mass of 10,000 kg and contains 500 birds, each weighing 200 g. When the birds are flying inside the container, their weight is still supported by the air pressure acting on the container's floor. Thus, the total weight measured by a scale at the bottom of the container remains 10,000 kg, as the lifting force generated by the birds does not alter the overall weight.

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Clausius2
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I have a little puzzle to share:
There is a container of a truck which mass is 10000 Kg. It is filled with 500 birds of 200 g each one.
What is the total weight measured of the container if birds are all flying inside?.
First I propose you to analyse the trivial case: the volume occuped by the birds is negligible compared with the volume of the container.
Second I propose you to analyse the non-trivial case: there is only one bird of 5000kg weighting inside. It remains steady aproximately at the center of the container, due to the movement of its wings that provide it lifting force. Now, what do you think?. Is the total weight measured by an steady scale at the floor more or less than 10000kg?.
There is air inside of the container with atmospheric pressure.
 
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