If an insect started flying inside a closed jar, would the jar weight stay the same?

The value read on the scale represents the mass of the jar, the insect and the air.

If the insect rose into the air, there would be fluctuation in the reading, which would depend on the direction in which the insect accelerated.

The fluctuation would tend to zero the longer the insect remained stationary in midair (i.e., if it did not accelerate in any direction, the reading would return the same value as when the insect was lying at the bottom of the jar).

It would increase slightly if the insect accelerated toward the lid, and decrease slightly if it accelerated toward the bottom.

Ideally you should be able to predict the absolute value of the instant-by-instant change (simplifying extremely), in the reading versus the total mass when the insect flies.