Ideal monotonicity conjecture for the polygamma Lehmer-mean difference

Let $i\ge0$ be an integer, $s,t\in\mathbb{R}$ with $s\ne t$, and $x> -\min\{s,t\}$. Define the generalized logarithmic mean $L_p(s,t;x)=L_p(x+s,x+t)$ and let

$$f_{i,p,s,t}(x)=(-1)^i\left[\psi^{(i)}(L_p(s,t;x))-\frac{1}{t-s}\int_s^t\psi^{(i)}(x+u)\operatorname{d}u\right].$$

Ideal monotonicity conjecture. The function $f_{i,p,s,t}$ is increasing with respect to $x$ if and only if $p\le -(i+1)$, and decreasing with respect to $x$ if and only if $p\ge -i$. This conjecture seeks the sharp necessary and sufficient parameter ranges extending the preceding sufficient monotonicity results; the source records those sufficient ranges but does not establish the asserted “if and only if” statement.