HOLP
The HOLP, short for Held Object's Last Position, represents the coordinates at which the object Mario held last was last rendered. The HOLP is not reset when changing courses or files, but is reset to the origin ([0, 0, 0]) when starting the game. When released, objects are re-activated and placed at the HOLP, which is usually in front of Mario. However, if the object was not rendered while being held by Mario, the HOLP would not update during that time, so the object won't appear in front of Mario. This can happen when Mario (and by extension the object he's holding) isn't rendered by the camera (which occurs while the camera is panned somewhere else or when he's at least 20,300 units away from the camera), when he he has his hat in his hand, or during Pause-Buffered Hitstun.
Effect on Objects
When an object is thrown, it will appear at the HOLP. When an object is dropped, it will appear at the HOLP laterally (i.e. share the HOLP's X and Z values) but will be at Mario's height (i.e. it will share his Y value). The exception to this is Bowser, who will always appear directly in front of Mario regardless of the HOLP.