Valuation conjectures for diskoids associated with abstract key polynomials

Let $b[?]$ be a valuation and $Q$ an abstract key polynomial (ABKP) for $b[?]$. For an associated diskoid $b[?](Q,b[?](Q))$, let $b[?]_{b[?](Q,b[?](Q))}$ denote the induced function on $b[?][X]$. An RT extension is the type of residually transcendental extension considered in the paper. Diskoid valuation conjectures. (C1) If $Q$ is an ABKP for $b[?]$, then $b[?]_{b[?](Q,b[?](Q))}$ is a valuation. (C2) If $b[?]$ is an RT extension, then there is a unique diskoid $D$ such that

$$b[?]=b[?]_D.$$

These conjectures seek, respectively, multiplicativity of the diskoid-induced function for abstract key polynomials and uniqueness of diskoid representation for RT extensions; the preceding counterexample shows that the analogous assertion does not hold for arbitrary diskoids.