Relationship between ordinary matrix product and hadamard product I would like to know is there any relationship between $(ABC)\circ D$ and $A(B\circ D)C$

Instead of general case, I would also like to know is there any special situational/ under any assumption that will make  $(ABC)\circ D$ and $A(B\circ D)C$ relate to each other.
I think it is not possible to perform any conversion. A counter-example:

Let's assume that A, B and C are matrices nxn of 1's. The rank of them is obviously one and also that of the product (ABC will be a nxn matrix of identical n^2 entries). After hadamrd-multiplying ABC by a random D, you can get anything, including rank $n$ matrices.

On the other hand, A (B * D) C is a rank one matrix. Therefore, little to do with it.
I think it is not possible to perform any conversion. A counter-example:

Let's assume that A, B and C are matrices nxn of 1's. The rank of them is obviously one and also that of the product (ABC will be a nxn matrix of identical n^2 entries). After hadamrd-multiplying ABC by a random D, you can get anything, including rank $n$ matrices.

On the other hand, A (B * D) C is a rank one matrix. Therefore, little to do with it.