The lower-bound conjecture for sufficiently complicated 2-complexes

Let $X$ be a 2-dimensional simplicial complex in which every 1-simplex is contained in at least two 2-simplices. The topological Turán number is denoted by $\operatorname{ex}_{\hom}(n,X)$. Lower-bound conjecture. There is a positive constant $c=c(X)$ such that

$$\operatorname{ex}_{\hom}(n,X)\geq cn^{5/2}.$$

This extends the known lower bound for surfaces to the proposed class of 2-dimensional complexes. The source explains that the conjecture is false without the condition on 1-simplices, while a proof for the stated class could follow from sufficiently many triangulations of each fixed $X$.