The measurable-function representation conjecture for QF-limits of binary functional structures

Let $\sigma$ be the signature consisting of a single binary functional symbol $f$. A QF-limit is the limit of a quantifier-free-convergent sequence of finite $\sigma$-structures. The space $\mathfrak{P}([0,1])$ denotes the space of probability measures on $[0,1]$.

Measurable-function representation conjecture. The limit of every QF-convergent sequence of finite $\sigma$-structures can be represented by a measurable function

$$w:[0,1]\times[0,1]\rightarrow\mathfrak{P}([0,1]).$$

Quantifier-free limits of general functional structures were open in the setting described by the source, even for unary functions; this conjecture proposes the natural limit object for structures with one binary function. The source does not state a resolution, so the conjecture remains open.