A two-sided inequality for the zeta function, digamma function, and cotangent

Let $B_1(x)$ be the first Bernoulli polynomial, let $\psi$ denote the digamma function, and let $\gamma=-\psi(1)$ be the Euler–Mascheroni constant. Define

$$\overline{b}:=\overline{B_1(\gamma)}=\gamma+\frac{1}{2},\qquad b:=B_1(\gamma)=\gamma-\frac{1}{2}.$$

For $0<x<1$, the inequality conjecture.

$$\pi\cot(\pi x)+x<\zeta(x)-\psi(x)<\pi\cot(\pi x)+\overline{b}x+b.$$

This inequality would give a simple comparison between the Euler–Riemann zeta function, the digamma function, and the cotangent on the unit interval. The source presents it as a problem arising from the study of a related theorem; no resolution is supplied here.