Sufficiency of the FLM inequality's necessary exponent condition

Let $a,b,c>0$ and suppose

$$c<1,\qquad a+b+2c=2,\qquad a,b\geq 1-2c.$$

For a sequence of polytopes $P_n\subset\mathbb{R}^n$, write $V_n$ and $\mathcal{F}_n$ for its vertices and facets, and let $r_nB_2^n\subset P_n\subset R_nB_2^n$. FLM exponent sufficiency conjecture. There exists a sequence of polytopes $P_n\subset\mathbb{R}^n$ such that

$$\log|V_n|\sim n^a,\qquad \log|\mathcal{F}_n|\sim n^b,\qquad \left(\frac{R_n}{r_n}\right)^2\sim n^{2c}.$$

The inequalities $a,b\geq 1-2c$ are necessary consequences of the classical FLM lower bounds and the relation $a+b+2c=2$; the conjecture asserts that these conditions are also sufficient. The claim concerns the unresolved tightness of the general FLM inequality outside the cases settled earlier in the paper.