The sulphide fractionation towards the core is a valid hypothesis. https://www.geochemicalperspectivesletters.org/article1506

The "lead paradoxa" express another inequality in earth's fractionation, if compared to for example meterorites.

To further constrain this, petrologists run some high-P/high-T experiments to see the behaviour of elements during the Earth differentiation. Some elements are said to be lithophile, meaning they will stay in the BSE, while other elements are considered siderophile, hence should have joined the metal phase during segregation of the core. The problem is that some experiments disagree with others, as the behaviour of a given element can change depending on the conditions. For instance, niobium and tantalum are considered lithophile, but can be siderophile in reduced conditions.

I'm afraid the bottom-line of all this is: there is no simple answer to this question...