We show that graph products of non trivial finite dimensional von Neumann algebras are strongly 1-bounded when the underlying *-algebra has vanishing first L2-Betti number. The proof uses a combination of the following two key ideas to obtain lower bounds on the Fuglede-Kadison determinant of matrix polynomials in a generating set: a notion called ''algebraic soficity'' for *-algebras allowing for the existence of Galois bounded microstates with asymptotically constant diagonals; a probabilistic construction of the authors of permutation models for graph independence over the diagonal.