On the Lusternik-Schnirelmann category of spaces with 2-dimensional fundamental group

The following inequality

$$\mathrm{cat}_{\mathrm{LS}} X\le\mathrm{cat}_{\mathrm{LS}} Y+\bigg\lceil\frac{hd(X)-r}{r+1}\bigg\rceil$$

holds for every locally trivial fibration $f:X\to Y$ between $ANE$ spaces which admits a section and has the $r$-connected fiber, where $hd(X)$ is the homotopical dimension of $X$. We apply this inequality to prove that

$$\mathrm{cat}_{\mathrm{LS}} X\le cd(\pi_1(X))+\bigg\lceil\frac{\dim X-1}{2}\bigg\rceil$$

for every complex $X$ with $cd(\pi_1(X))\le 2$, where $cd(\pi_1(X))$ denotes the cohomological dimension of the fundamental group of $X$.