Theory of Storage , Convenience Yield and Inventories

CFA Level II Corporate Finance
#1

I was doing one of the mocks that asked to choose the correct statement

This was one the statements that I thought was correct- “theory of storage, which predicts a direct relationship between the level of inventories and the convenience yield.”
However, apparently, it is wrong because of the following:
The theory of storage predicts an inverse relationship between the level of inventories and the convenience yield.

I thought that when we want to own something badly rather than going long on it with a future contract, that the current level of inventories would increase because we are owning the asset no?

#2

The convenience yield is the benefit of having access to a commodity when inventories are low or are expected to be low. So it’s the benefit of having a buffer in case there is a supply shock.

Therefore if inventories are high then the convenience yield will be low and when inventories are low the convenience yield will be high.

#4

I think I am confused here.
The way I interpreted this was that when the convenience yield is high, you buy the commodity and store it in your inventory which makes it high. So arent they directly related?