The flag-complex Turán lower-bound conjecture

Let $\Delta$ be a flag complex with $\beta_k(\Delta)=a>0$, and let $T$ be any $k$-dimensional Turán complex satisfying $\beta_k(T)\leq a$.

Turán lower-bound conjecture. The face vector of $\Delta$ is componentwise at least that of $T$:

$$f(\Delta)\geq f(T).$$

This is presented as a consequence of the preceding conjecture and would provide a quantitative refinement of Meshulam's theorem for flag complexes in all dimensions.