Realizing Cuntz classes in commutative subalgebras

Let $A$ be a unital, simple C*-algebra of stable rank one. Does there exist a commutative sub-C*-algebra $C(X)\subseteq A$ such that for every lower-semicontinuous function $f\colon\mathrm{QT}_1(A)\to[0,1]$ there exists an open subset $U\subseteq X$ such that $f(\tau)=\mu_\tau(U)$ for $\tau\in\mathrm{QT}_1(A)$ ? Here, $\mathrm{QT}_1(A)$ denotes the Choquet simplex of normalized $2$ -quasitraces on $A$ (if $A$ is exact, then this is just the Choquet simplex of tracial states on $A$ ), and $\mu_\tau$ denotes the probability measure on $X$ induced by the restriction of $\tau$ to $C(X)$ .

More specifically, one may ask if this is always the case for a Cartan subalgebra of $A$ .

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