We show that a graph product of tracial von Neumann algebras is strongly $1$-bounded if the first $\ell^2$-Betti number vanishes for an associated dense $*$-subalgebra.  Graph products of tracial von Neumann algebras were studied by Caspers and Fima, and generalize Green's graph product of groups.  Given groups $G_v$ for each vertex of a graph $\Gamma$, the graph product is the free product modulo the relations that $G_v$ and $G_w$ commute when $v \sim w$; for von Neumann algebras, graph products are described by a certain moment relation.  In our paper, the crux of the argument is a generalization to tracial von Neumann algebras of the statement that soficity of groups is preserved by graph products.  We replace soficity for groups with a more general notion of algebraic soficity for a $*$-algebra $A$, which means the existence of certain approximations for the generators of $A$ by matrices with algebraic integer entries and approximately constant diagonal.  We show algebraic soficity is preserved under graph products through a random permutation construction, inspired by previous work of Charlesworth and Collins as well as Au-C{\'e}bron-Dahlqvist-Gabriel-Male.  In particular, this gives a new probabilistic proof of Ciobanu-Holt-Rees's result that soficity of groups is preserved by graph products.

This is based on joint work with Rolando de Santiago, Ben Hayes, Srivatsav Kunnawalkam Elayavalli, Brent Nelson.