Strong 1-boundedness characterized by vanishing first $L^2$-Betti number

Let $(M,\tau)$ be a tracial von Neumann algebra, and let $A$ be a weak$^{*}$-dense, finitely presented, algebraically sofic, unital $*$-subalgebra of $M$. Write $\beta^{1}_{(2)}(A,\tau)$ for the first $L^2$-Betti number of $A$ with respect to $\tau$. Strong 1-boundedness conjecture. The algebra $(M,\tau)$ is strongly $1$-bounded if and only if

$$\beta^{1}_{(2)}(A,\tau)=0.$$

This conjecture proposes a characterization of strong $1$-boundedness using the first $L^2$-Betti number for algebraically sofic, finitely presented dense subalgebras. Its status is not resolved in the supplied source context.