Merino-Welsh inequalities for matroids with controlled lattices of cyclic flats

Beke, Csáji, Csikvári, and Pituk showed that the Merino–Welsh quotient $\mathrm{\Phi }(M)=T_M(1,1{)}^2\mathrm{/}(T_M(2,0)T_M(0,2))$ can be arbitrarily large, so the multiplicative Merino-Welsh inequality fails for matroids in general. We show that $\mathrm{\Phi }$ is uniformly bounded on the class of matroids whose cyclic-flat lattice avoids any fixed finite poset $P$ as an induced subposet. We further prove $\mathrm{\Phi }(M)\le 1$ for matroids of cyclic width at most $7$, cyclic height at most $6$, and for loop- and coloop-free $4$-paving or $4$-copaving matroids.