The flag-complex canonical-representation lower-bound conjecture

Let $\Delta$ be a flag complex with $\beta_{k-1}(\Delta)=a>0$. Suppose that

$$a=\binom{a_k}{k}_{\!k}+\binom{a_{k-1}}{k-1}_{\!k-1}+\dots+\binom{a_{k-s}}{k-s}_{\!k-s}$$

is the $(k,k)$-canonical representation of $a$.

Canonical-representation lower-bound conjecture. For every $i\geq 0$,

$$f_{i-1}(\Delta)\geq\binom{a_k+k}{i}_{\!k}+\binom{a_{k-1}+k-1}{i-1}_{\!k-1}+\dots+\binom{a_{k-s}+k-s}{i-s}_{\!k-s}.$$

This conjecture would give a quantitative refinement of Meshulam's theorem for flag complexes in all dimensions and would imply the subsequent Turán-complex lower-bound conjecture.