Consider the following variant of Freiling's Axiom of Symmetry, $\mathsf{AS}$, which will be denoted $A_{< 2^{\aleph_0}}$:

- given any function $f$ from $\mathbb{R}$ into the families of of subsets of $\mathbb{R}$ of size $< 2^{\aleph_0}$ there are $x_1,x_2 \in \mathbb{R}$ such that $x_1 \notin f(x_2)$ and $x_2 \notin f(x_1)$.

Does $\mathsf{ZF} + A_{<2^{\aleph_0}} + \mathsf{LM}$ imply $\neg \mathsf{WCH}$? If not, what is the current state of research in discovering models of $\mathsf{ZF}$ in which $\mathsf{LM} +\lnot \mathsf{WCH}$ holds?

Here

- $\mathsf{LM}$ is the statement that every set of reals is Lebesgue measurable,
- $\mathsf{WCH}$ is the statement that every uncountable subset of $\mathbb R$ can be put into 1-1 correspondence with $\mathbb R$.

For background to this question, in his 3 August 2011 FOM post to Timothy Chow, Ali Enayat stated that

$\mathsf{ZF} + \mathsf{AS} + \mathsf{LM} + \mathsf{WCH}$ holds in Solovay's model since in Solovay's model every uncountable subset of $\mathbb{R}$ has a perfect subset.